Chapter 5: Electrostatics and Magnetism

gravitational force (In the electrostatic force equation, the force magnitude is proportional to the charge magnitudes, and this is similar to the proportional relationship between gravitational force and mass. In both equations, the force magnitude is inversely proportional to the square of the distance of separation.)

A close examination of Coulomb's law, Fe = kq₁q₂/r², reveals that it is remarkably similar in form to the equation for ______ ______ = Gm₁m₂/r².

A (Coulomb's law, Fe = kq₁q₂/r², states that the force between two charges varies as the inverse of the square of the distance between them. Therefore, if the distance is doubled, the square of the distance is quadrupled, and the force is reduced to one-fourth of what it was originally.)

A positive charge is attracted to a negative charge a certain distance away. The charges are then moved so that they are separated by twice the distance. How has the force of attraction changed between them? A.) 1/4 B.) 1/2 C.) 2 D.) 4

translational (For a dipole at some angle in an external electric field, there will be translational equilibrium, but not rotational equilibrium. This is because the forces are opposite directions, but the torques are in the same direction--clockwise for both.)

An electric field will not induce any ______ motion in the dipole regardless of its orientation with respect to the electric field vector.

E (From the equation for electrical potential (V = U/q = kQq/r), we can see that no work is done when moving a test charge q from one point on an eqipotenial line to another. Work will be done in moving a test charge q from one line to another, but the work depends only on the potential difference of the two lines and not on the pathway taken between them; thus any of the paths shown would require the same amount of work in moving the electron from a to b. Note that because the source charge is positive, point b is at a lower electrical potential than point a. However, because the test charge is negative, the electrical potential energy is higher at point b than point a. This should make sense: the electron will have to gain energy to be moved farther away from a positive source charge.)

An electron goes from point a to point b in the vicinity of a very large positive charge. Which path requires the least work to get the electron charge from a to b? A.) Green B.) Purple C.) Pink D.) No work is required for any path. E.) The same amount of work is required for all paths.

north (Field lines exit the north pole and enter the south pole.)

Bar magnets are ferromagnetic materials with a north and south pole. Which end of the magnet do field lines exit from?

positive (Thus, if the source charge is positive, then the test charge would experience a repulsive force and would accelerate away from the positive source charge. On the other hand, if the source charge is negative, then the test charge would experience an attractive force and would accelerate toward the negative source charge.)

By convention, the direction of the electric field vector is given as the direction that a(n) ______ test charge would move in the presence of the source charge.

decrease (The electrical potential energy of a system will decrease when two like charges move apart of when two opposite charges move toward each other.)

Charges, if allowed, will move spontaneously in whatever direction results in a(n) (DECREASE / INCREASE) in electrical potential energy.

permittivity of free space

Coulomb's constant, k, (also called the electrostatic constant) is a number that depends on the units used in the equation. In SI units, k = 1/4πε₀ = 8.99 x 10⁹ N∙m²/C², where ε₀ represents the ______ ______ ______ ______, 8.85 x 10⁻¹² C²/N∙m².

E (FB = qvBsinθ; Since the particle is a neutron and q=0, there is no magnetic force.)

Determine the direction of the magnetic force (FB) on a neutron with velocity (v) out of the page and a magnetic field (B) pointing to the right of the page. A.) into the page B.) out of the page C.) up the page D.) down the page E.) none

E (The charge is moving with a velocity antiparallel to the magnetic field vector and will thus experience no magnetic force. FB = qvBsinθ → sin180 = 0.)

Determine the direction of the magnetic force (FB) on a proton with velocity (v) into the page and a magnetic field (B) pointing out of the page. A.) into the page B.) out of the page C.) up the page D.) down the page E.) none

C

Determine the direction of the magnetic force (FB) on a proton with velocity (v) to the right and a magnetic field (B) pointing into the page. A.) into the page B.) out of the page C.) up the page D.) down the page E.) none

A

Determine the direction of the magnetic force (FB) on an electron with velocity (v) up the page and a magnetic field (B) pointing to the left. A.) into the page B.) out of the page C.) down the page D.) up the page E.) none

increase (A decrease in potential energy indicates that the system has become more stable. Keep in mind that negative numbers with larger absolute values are more negative, and represent a decrease in value from negative numbers with smaller absolute values. That is, -4 > -7 even though |-4| < |-7|.)

Does a change in electrical potential energy from -4 J to -7 J decrease, increase, or not change the stability of a system?

C (Field lines are imaginary lines that represent how a positive test charge would move in the presence of the source charge. They point away from a positive charge and point toward a negative charge.)

Electric field vectors can be represented as field lines that radiate ______ from positive source charges and radiate ______ to negative source charges. A.) parallel, perpendicular B.) perpendicular, parallel C.) outward, inward D.) inward, outward

source, test, qE

Electric fields are produced by ______ charges (Q). When a(n) ______ charge (q) is placed in an electric field (E), it will experience an electrostatic force (Fe) equal to ______.

positive (The sign of the electrical potential is determined by the sign of the source charge: V = kQ/r. For a collection of charges, the total electrical potential at a point in space is the scalar sum of the electrical potential due to each charge.)

Electrical potential (V) is a scalar quantity. What is its sign for a positive source charge?

positive (Electrical potential energy is given by the equation: U=kQq/r. Like charges will cancel to give a positive potential energy. To conceptualize this, consider two positive charges. Because like charges repel each other, the closer they are, the more work (electrical potential energy) will be needed to move the test charge from infinity to the electric field surrounding the source charge.)

Electrical potential energy is form of potential energy that is dependent on the relative position of one charge with respect to another charge or to a collection of charges. If the charges are like charges (both positive or both negative), will the potential energy be positive or negative?

negative (Electrical potential energy is given by the equation: U=kQq/r. Opposite charges will give a negative potential energy. To conceptualize this, consider a stationary negative source charge and a positive test charge that can be moved. Because these two charges are unlike, they will exert attractive forces between them. Therefore, the closer they are to each other, the more stable they will be and the less work (electrical potential energy) will be needed to move the test charge from infinity to the electric field surrounding the source charge.)

Electrical potential energy is form of potential energy that is dependent on the relative position of one charge with respect to another charge or to a collection of charges. If the charges are opposite (one positive and the other negative), will the potential energy be positive or negative?

B

Equipotential lines are always ______ to electric field lines. A.) Parallel B.) Perpendicular C.) Tangentinal D.) Equivalent

magnetic field, permeability of free space

For an infinitely long and straight current-carrying wire, we can calculate the magnitude of the magnetic field produced by the current I in the wire at a perpendicular distance, r, from the wire as: B = μ₀I/2πr, where B is the ______ ______ at a distance r from the wire, μ₀ is the ______ ______ ______ ______ (4π x 10⁻⁷ T∙m/A), and I is the current.

kQ/r^2 (The magnitude of the electric field can also be expressed as E = Fe/q, which does not require the value of the source charge.)

Give the equation for the magnitude of an electric field (E) in terms of the electrostatic constant (k), the source charge magnitude (Q), and the distance between the charges (r). HINT: this can be obtained by dividing Coulomb's law by the magnitude of the test charge.

Fe/q (The magnitude of the electric field can also be expressed as E = kQ/r², which does not require the presence of a test charge to calculate the field strength.)

Give the equation for the magnitude of the electric field (E) in terms of the electrostatic force (Fe) felt by a test charge (q). HINT: this can be obtained by dividing Coulomb's law by the magnitude of the test charge.

D (A charge will move in such a way to minimize its potential energy. Placing a charge at a point of zero electrical potential does not indicate that there is zero potential difference, so the charge may or may not move--and if it moves, it may move toward or away from the source charge depending on the sign of the source charge and test charge.)

How will a charge that is placed at a point of zero electrical potential move relative to a source charge? A.) toward the source charge B.) away from the source charge C.) it will be stationary D.) Not enough information given

pEsintheta (where p is the magnitude of the dipole moment (p=qd), E is the magnitude of the uniform external electric field, and θ is the angle the dipole moment makes with the electric field)

In an external electric field, an electric dipole moment (p) will experience a net torque of τ = ______ until it is aligned with the electric field vector (E).

ground

In electrostatics, a(n) _______ refers to a means of returning charge to the earth. For example, when you shuffle your feet across the carpet, negatively charged particles are transferred from the carpet to your feet, and these charges spread out over the total surface of your body. The shock that occurs when your hand get close enough to a metal doorknob allows that excess charge to jump from your fingers to the knob, which acts as the earth in this scenario.

centripetal force

Point charges may undergo uniform circular motion in a uniform magnetic field wherein the ______ ______ is the magnetic force acting on the point charge.

more (Lower humidity makes it easier for charge to become and remain separated.)

Static charge buildup or static electricity is (LESS / MORE) significant in drier air.

C (To determine the direction, use the right-hand rule. Your thumb should point up the page in the direction v. Your fingers should point into the page in the direction of B. Protons are positively charged; thus the force, FB, is in the direction of your palm, which is to the left with a magnitude of FB = qvBsinθ = (1.60 x 10⁻¹⁹ C)(15 m/s)(3.0 T)sin90° = 7.2 x 10⁻¹⁸ N. Note that v and FB will always be perpendicular to each other; this implies that uniform circular motion will occur in this field, with FB pointing radially inward toward the center of the circle.)

Suppose a proton is moving with a velocity of 15 m/s toward the top of the top of the page through a uniform magnetic field of 3.0 T directed into the page, as shown in the figure. What is the direction of the magnetic force on the proton? A.) into the page B.) out of the page C.) left D.) right E.) None

A (Use the right-hand rule to determine the direction of the magnetic field within and outside of the loop. Align you right thumb with the current at any point in the loop. When you encircle the wire with the remaining fingers of your right hand, your fingers should point in to the page within the loop and out of the page outside of the loop.)

Suppose a wire is formed in a loop that carries a current of 0.25 A in a clockwise direction, as shown in the figure. Determine the direction of the magnetic field produced by this loop within the loop and outside the loop. A.) into the page, out of the page B.) out of the page, into the page C.) into the page, into the page D.) out of the page, out of the page

C (To determine the magnitude of the magnetic field at the center, use the equation for a loop of wire: B = μ₀I/2r = [(4π x 10⁻⁷ T∙m/A)(0.25 A)] / (2 x 0.5 m) ~ 3.14 x 10⁻⁷ T = 3.14 x 10⁻³ gauss.)

Suppose a wire is formed in a loop that carries a current of 0.25 A in a clockwise direction, as shown in the figure. If the loop has a diameter of 1 m, which of the following correctly gives the magnetic field at the center of the loop? A.) [(4π x 10⁻⁷ T∙m/A)(0.25 A)] / (2 x π x 0.5 m) B.) [(4π x 10⁻⁷ T∙m/A)(0.25 A)] / (2 x π x 1.0 m) C.) [(4π x 10⁻⁷ T∙m/A)(0.25 A)] / (2 x 0.5 m) D.) [(4π x 10⁻⁷ T∙m/A)(0.25 A)] / (2 x 1.0 m)

electric field (E)

The ______ ______ is the ratio of the force that is exerted on a test charge to the magnitude of that charge = Fe/q = kQ/r².

Lorentz force

The ______ ______ is the sum of the electrostatic and magnetic forces acting on a body.

stronger

The denser the field lines, the (WEAKER / STRONG / EQUAL) the electric field.

decrease

The electrical potential energy of a system will (DECREASE / INCREASE) when two like charges move apart.

increase

The electrical potential energy of a system will (DECREASE / INCREASE) when two like charges move toward each other.

increase

The electrical potential energy of a system will (DECREASE / INCREASE) when two opposite charges move apart.

decrease

The electrical potential energy of a system will (DECREASE / INCREASE) when two opposite charges move toward each other.

B (The equations for a long, straight wire (B = μ₀I/2πr) and a circular loop of wire (B = μ₀I/2r) are quite similar--the obvious difference being that the equation for the magnetic field at the center of the circular loop of wire does not include the constant π. The less obvious difference is that the first expression gives the magnitude of the magnetic field at any perpendicular distance, r, from the current-carrying wire, while the second expression gives the magnitude of the magnetic field only at the center of the circular loop of current-carrying wire with radius r.)

The equation B = μ₀I/2r gives the magnetic field produced by the current in a: A.) long, straight wire B.) circular loop of wire

A (The equation demonstrates an inverse relationship between the magnitude of the magnetic field and the distance from the current. Straight wires create magnetic fields in the shape of concentric rings. To determine the direction of the field vectors, use a right-hand.)

The equation B = μ₀I/2πr gives the magnetic field produced by the current in a: A.) long, straight wire B.) circular loop of wire

perpendicular bisector, zero

The plane that lies halfway between +q and -q is an important equipotential line called the ______ ______ of the dipole. Because the angle between this plane and the dipole axis is 90° (and cos90°=0), the electrical potential at any point along this plane is _____.

gauss

The size of the tesla unit is quite large, so small magnetic fields are sometimes measured in units of 1 T = 10⁴ ______.

electric field (E = Fe/q = kQ/r²), magnetic field (B = μ₀I/2πr for a straight wire and B = μ₀I/2r for a loop of wire), magnetic force (FB = qvBsinθ on a moving point charge and FB = ILBsinθ on a current-carrying wire)

To create a nonzero ______ ______, one needs a charge. To create a(n) _____ _____, one needs a charge that must also be moving. Finally, to create a(n) ______ ______, one needs an external electric field acting on a charge moving any direction except parallel or antiparallel to the external field.

False (It is true that opposite charges will have negative potential energy, but this energy will become increasingly negative as the charges are brought closer and closer together. Increasingly negative numbers are actually decreasing values because they are moving farther to the left of 0 on the number line. This decrease in energy represents an increase in stability. Conceptually, when two opposite charges are closer together, it will take less work (electrical potential energy) to move the test charge to the electric field surrounding a test charge.)

True or False: A stationary negative source charge and positive test charge will have increasingly negative electrical potential energy as they are brought further apart from each other and will thus increase in stability.

True (As like charges, these will exert repulsive forces, and the potential energy of the systems will be positive. Because like charges repel each other, the closer they are to each other, the less stable they will be. Conversely, the like charges will become more stable the farther apart they move because the magnitude of the electrical potential energy becomes a smaller and smaller positive number.)

True or False: A stationary positive source charge and positive test charge will decrease in positive electrical potential energy as they are brought further apart from each other and will thus increase in stability.

True (Every electric charge sets up a surrounding electric field, just like every mass creates a gravitational field.)

True or False: Every charge generates an electric field, which can exert force on other charges.

True (When a charge moves in a magnetic field, the magnetic force can be calculated from the following equation: FB = qvBsinθ. Remember that sin0° and sin180° equal zero. This means that any charge moving parallel or antiparallel to the direction of the magnetic field will experience no form from the magnetic field.)

True or False: External magnetic fields exert forces on charges moving in any direction except parallel or antiparallel to the field.

False (Even if there is no test charge, q, we can still calculate the electrical potential of a point in space in an electric field as long as we know the magnitude of the source charge and the distance from the source charge to the point in space in the field. By dividing U=kQq/r by q, we get V = kQ/r.)

True or False: If there is no test charge, q, it is impossible to determine the electrical potential because V = U/q.

True (Because magnetic field lines are circular and point from north to south, it is impossible to have a monopole magnet.)

True or False: It is impossible to have a monopole magnet.

False (No work is done when a charge moves from a point on an equipotential line to another point on the same equipotential line. However, work WILL be done when a charge is moved from one equipotential line to another; the work is independent of the pathway taken between the lines.)

True or False: No work is done when a charge moves from one equipotential line to another.

False (The atoms of both paramagnetic and ferromagnetic materials have unpaired electrons, so these atoms do have a net magnetic dipole moment, but the atoms in these materials are usually randomly oriented so that the material itself creates no net magnetic field.)

True or False: Paramagnetic and ferromagnetic materials create a magnetic field of weak and strong strengths, respectively.

True (The "plus" end of a battery is the high-potential end, and the "minus" end of a battery is the low-potential end. Positive charges will spontaneously move in the direction that decreases their electrical potential (negative voltage), whereas negative charges will spontaneously move in the direction that increases their electrical potential (positive voltage)--yet, in both cases, the electrical potential energy is decreasing.)

True or False: Positive charges in a battery move from + to - while negative charge moves from - to +.

False (Unlike the electrostatic force, the electric field is unrelated to test charge but is still related to distance by an inverse square relationship defined by E = kQ/r². Since it is the source charge that creates the electric field--not the test charge--we cannot use the equation E = Fe/q to determine a relationship.)

True or False: The electric field is directly related to the charge and related to the distance by an inverse square relationship.

True (The ratio between these forces can be calculated by dividing their magnitudes: Fe/Fg = [kq₁q₂/r²]/[Gm₁m₂/r²] = kq₁q₂/Gm₁m₂ = [(8.99 x 10⁹ N∙m²/C²)(1.60 x 10⁻¹⁹ C)(1.60 x 10⁻¹⁹ C)]/[(6.67 x 10⁻¹¹ N∙m²/C²)(1.67 x 10²⁷ kg)(9.11 x 10⁻³¹ kg)] ~ 1.6/6.67 x 10⁴⁰ ~ 2.5 x 10³⁹. Thus, the electrostatic attraction between an electron and proton is stronger than the gravitational attraction by a factor of almost 10⁴⁰. Make note of the values for constants, masses, and charges because they may be useful on Test Day.)

True or False: The electrostatic attraction between an electron and proton is stronger than the gravitation attraction between the two.

True (This is supported by Coulomb's law: Fe = kq₁q₂/r².)

True or False: The electrostatic force is directly related to the charge and related to the distance by an inverse square relationship.

False (The force of gravity is always an attractive force. This contrasts the electrostatic force which may be repulsive or attractive depending on the signs of the charges that are interacting.)

True or False: The force of gravity can be an attractive or repulsive force depending on mass and charge.

False (The electrical POTENTIAL (V=[kqd/r²]cosθ) is zero because the angle between the perpendicular bisector and dipole axis is 90° and cos90°=0. However, the electric FIELD is not zero and is approximated as E = 1/4πε₀ x p/r³.)

True or False: The magnitude of the electric field on the perpendicular bisector of a dipole is zero.

True (An equipotential line is a line on which the potential at every point is the same. Thus, no work is done when a charge moves from a point on an equipotential line to another point on the same equipotential line.)

True or False: The potential difference between any two points on an equipotential line is zero.

True (Electrical potential energy is measured in joules and is equal to U=kQq/r, while electrical potential (V = U/q = kQ/r) and potential difference (ΔV = Vb - Va = Wab/q) are measured in volts.)

True or False: The units of electrical potential energy and electrical potential are different.

electric dipole

Two charges of opposite sign separated by a fixed distance d generate a(n) ______ ______.

coulomb-meter (p=qd)

What are the SI units of the dipole moment (p)?

Fe = kq1q2/r^2 (where Fe is the magnitude of the electrostatic force, k is Coulomb's constant, q₁ and q₂ are the magnitudes of the two charges, and r is the distance between the charges)

What is Coulomb's law?

tesla (1 T = 1 N∙s/m∙C)

What is the SI unit for magnetic field strength?

coulomb (1 C = 1 A ⋅ s)

What is the SI unit of charge?

B (In the absence of an electric field, the dipole axis can assume any random orientation, choice D, but this dipole is exposed to an external electric field. When the electric dipole is placed in a uniform external electric field, each of the equal and opposite charges of the dipole will experience a force exerted on it by the field. Because the charges are equal and opposite, the forces acting on the charges will also be equal in magnitude and opposite in direction, resulting in a situation of translational equilibrium, eliminating choice A. There will, however, be a net torque about the center of the dipole axis = pEsinθ that causes the dipole to reorient itself so that its dipole moment aligns with the electric field as in choice B.)

What is the behavior of an electric dipole when exposed to a uniform external electric field? A.) undergo translation with the field B.) rotate such that its dipole moment aligns with the field C.) rotate such that its dipole moment is perpendicular with the field D.) assumes any arndom orientation

B (The electric field would be 0 because the two test charges are the same. In this case, the fields exerted by each charge at the midpoint will cancel out and there will be no electric field: E = Fe/q.)

What is the electric field midway between two negative charges in isolation? A.) -2 N/C B.) 0 N/C C.) 1 N/C D.) 2 N/C

(ΔV = Vb - Va=) Wab/q

What is the equation for potential difference in terms of Wab, the work needed to move a test charge q through an electric field form point a to b?

kqd/r^2 costheta (We know that for a collection of charges, the electrical potential P is the scalar sum of the potentials due to each charge at that point: V = kq/r₁ - kq/r₂ = kq(r₂-r₁)/r₁r₂. For points in space relatively distant from the dipole (compared to d), the product of r₁ and r₂ is approximately equal to the square of r, and r₁ - r₂ is approximately equal to dcosθ. When we plug these approximations into the previous equation, we get: V = [kqd/r²]cosθ.)

What is the equation for the electrical potential (V) at point P near a dipole?

qvBsintheta (where q is the charge, v is the magnitude of its velocity, B is the magnitude of the magnetic field, and θ is the smallest angle between the velocity vector v and the magnetic field vector B)

What is the equation for the magnetic force (FB) experienced by a charge moving in a magnetic field?

ILBsintheta (where I is the current, L is the length of the wire in the field, B is the magnitude of the magnetic field, and θ is the angle between L and B)

What is the equation for the magnetic force (FB) experienced by a straight current-carrying wire in a magnetic field?

kQ/r (This is not be to confused with the electrical potential energy, U, which equals kQq/r.)

What is the formula for electrical potential (V)?

kQq/r (This is NOT to be confused with the electrical potential V, which equals kQ/r.)

What is the formula for electrical potential energy (U)?

magnetic force (FB), velocity (v), magnetic field (B)

When a charge moves in a magnetic field, a(n) ______ _____ may be exerted on it, the magnitude of which is given by = qvBsinθ, where q is the charge, v is the magnitude of its velocity, B is the magnitude of the magnetic field, and θ is the smallest angle between the _____ vector and the ______ _____ vector.

A (The electrons will experience the greater acceleration because they are subject to the same force as the protons but have a significantly smaller mass.)

When placed one meter apart from each other, which will experience a greater acceleration? A.) one coulomb of electrons B.) one coulomb of protons C.) one coulomb of neutrons D.) one coulomb of electrons will experience the same acceleration as one coulomb of protons

D (Any magnet or moving charge, whether a single electron traveling through space or a current through a conductive material, creates a magnetic field.)

Which of the following will create a magnetic field? I. an electron moving through space II. current through a copper wire III. a permanent magnet A.) III only B.) I and II only C.) II and III only D.) I, II, and III

B (We need not determine the actual values of the magnetic fields in these two cases and can compare the two equations instead. The magnetic field created by the current-carrying wire is given by B = μ₀I/2πr; the magnetic field created by the loop of wire is given by B = μ₀I/2r. μ₀, I, and r are the same in both equations. Therefore, the magnetic field at the center of the loop must be larger because the denominator in that equation does not include π.)

Which would experience a larger magnetic field: (A) an object placed five meters to the left of a current carrying wire, or (B) an object placed at the center of a circle with a radius of five meters? (Note: Assume the current is constant; μ₀=4π x 10⁻⁷ T∙m/A)

Electrical potential energy

______ ______ ______ is the amount of work in joules required to bring a test charge from infinitely far away to a given position in space in an electric field surrounding a source charge.

Field lines

______ ______ are imaginary lines that represent how a positive test charge would move in the presence of the source charge. They are used to represent the electric field vectors for a charge.

Equipotential lines

______ ______ designate the set of points around a source charge or multiple source charges that have the same electrical potential.

Coulomb's law

______ ______ gives the magnitude of the electrostatic force vector between two charges. The force vector always points along the line connecting the centers of the two charges and the direction of the force may be obtained by remembering that unlike charges attract and like charges repel.

Potential difference (voltage)

______ ______ is the change in electrical potential that accompanies the movement of a test charge from one position to another.

Electrical potential

______ ______ is the ratio of the work done to move a test charge from infinity to a point in an electric field surrounding a source charge divided by the magnitude of the test charge. In other words, the electrical potential energy per unit charge in volts (J/C): V = U/q = kQ/r.

Conductors

______ allow the free and uniform passage of electrons when charged (often used in circuits or electrochemical cells; usually metals or ionic solutions).

Electrostatics

______ is the study of stationary charges and the forces that are created by and which act upon these charges.

Diamagnetic

______ materials possess no unpaired electrons and are slightly repelled by a magnet (e.g. wood, plastics, water, glass, skin).

Ferromagnetic

______ materials possess some unpaired electrons and become strongly magnetic in an external magnetic field (e.g. iron, nickel, cobalt).

Paramagnetic

______ materials possess some unpaired electrons and become weakly magnetic in an external magnetic field (e.g. aluminum, copper, gold).

Insulators

______ resist the movement of charge and will have localized areas of charge that do not distribute over the surface of the material or to another neutral object (e.g. nonmetals, dielectric materials in capacitors, prevention of grounding in isolated electrostatic experiments).

Positive

______ test charges will move in the direction of the field lines.

Negative

______ test charges will move in the direction opposite of the field lines.

Positive (Charges, if allowed, will move spontaneously in whatever direction results in a decrease in electrical potential energy. For a positive test charge, this means moving from a position of higher electrical potential to a position of lower electrical potential. The voltage, ΔV = Vb - Va, is negative in this case, because q is positive (for a positive test charge), Wab must also be negative, which represents a decrease in electrical potential energy.)

______ test charges will move spontaneously from high potential to low potential.

Negative (Charges, if allowed, will move spontaneously in whatever direction results in a decrease in electrical potential energy. For a negative test charge, this means moving from a position of lower electrical potential energy to a position of higher electrical potential. The voltage, ΔV = Vb - Va, is positive in this case; because q is negative (for a negative test charge), Wab must also be negative, which represents a decrease in electical potential energy.)

______ test charges will move spontaneously from low potential to high potential.