Physics Chapter 17: Electrical Energy and Current

Electrical Potential Energy

Potential energy associated with a charge due to its position in an electric field.

capacitance

The ability of a conductor to store energy in the form of electrically separated charges

Drift Velocity

The net velocity of a charge carrier moving in an electric field.

Resistance

The opposition presented to electric current by a material or device.

Electric Current

The rate at which electric charges pass through a given area

Resistors can be used to control the amount of current in a conductor

- One way to change the current in a conductor is to change the potential difference across the ends of the conductor. - According to the definition of resistance, if ∆V remains constant, current decreases when resistance increases. Thus, the current in a wire can be decreased by replacing the wire with one of higher resistance. - The same effect can be accomplished by making the wire longer or by connecting a resistor to the wire. - A *resistor* is a simple electrical element that provides a specified resistance.

Potential difference is a change in electric potential

- The potential difference between two points can be expressed as follows: ΔV = ΔPEelectric/q - Potential difference is a measure of the difference in the electrical potential energy between two positions in space divided by the charge. - The SI unit for potential difference (and for electrical potential) is the *volt, V* which is equivalent to *one joule per coulomb* - As a 1 C charge moves through a potential difference of 1 V, the charge gains 1 J of energy.

Potential Difference Between a Point at Infinity and a Point Near a Point Charge

ΔV = kc(q/r) potential difference = (Coulomb constant) x (value of the point charge / distance to the point charge)

Potential Difference (Equation)

ΔV = ΔPEelectric/q potential difference = change in electric potential energy / electric charge Units: Volts (V) = J/C PEelectric = Joules (J) q = charge (C)

Electrical potential energy can be associated with a charge in a uniform field

- Consider a positive charge in a uniform electric field - Assume the charge is displaced at a constant velocity in the same direction as the electric field - There is a change in electrical potential energy associated with the charge's new position in the electric field. - The change in the electrical potential energy depends on the charge, q, as well as the strength of the electric field, E, and the displacement, d. It can be written as follows: ΔPEelectric = -qEd - The negative sign indicates that the electrical potential energy will increase if the charge is negative and decrease if the charge is positive

Electric Potential Energy vs. Electric Potential and Potential Difference

- Electrical potential energy is a quantity of energy, with units in joules - Electric potential and potential difference are both measures of energy per unit charge (measured in units of volts) - Potential difference describes the change in energy per unit charge

Potential Difference in a Uniform Electric Field (Equation)

∆V = -Ed potential difference = -(magnitude of the electric field x displacement)

Resistance is constant over a range of potential differences

- For many materials, including most metals, experiments show that the resistance is constant over a wide range of applied potential differences. - This statement, known as Ohm's law, is named for Georg Simon Ohm (1789-1854), who was the first to conduct a systematic study of electrical resistance. - Mathematically, Ohm's law is stated as follows: V/I = constant - As seen by comparing the definition of resistance with Ohm's law, the constant of proportionality in the Ohm's law equation is resistance. - It is also common practice to express Ohm's law as V = IR

Ohm's law does not hold for all materials

- Materials that have a constant resistance over a wide range of potential differences are said to be ohmic. - A graph of current versus potential difference for an ohmic material is linear. This is because the slop of such a graph (I/∆V) is inversely proportional to resistance. - When resistance is constant, the current is proportional to the potential difference and the resulting graph is a straight line. - Materials that do not function according to Ohm's law are said to be non-ohmic. - In a graph of current versus potential difference for a non-ohmic material the slope is not constant because resistance varies. Hence, the resulting graph is nonlinear. - One common semiconducting device that is non-ohmic is the diode.

Electrical potential energy is a component of mechanical energy

- Mechanical energy is conserved as long as friction and radiation are not present - As with gravitational and elastic potential energy, electrical potential energy can be included in the expression for mechanical energy. - If a gravitational, elastic, and electric force are all acting on an object, the mechanical energy can be written as follows: ME = KE + PEgrav + PEelastic + PEelectric

Energy and Capacitors

A charged capacitor stores electrical potential energy because it requires work to move charges through a circuit to the opposite plates of a capacitor. The work done on these charges is a measure of the transfer of energy.

Discharging a capacitor releases its charge

- Once a capacitor is charged, the battery or other source of potential difference that charged it can be removed from the circuit. - The two plates of the capacitor will remain charged unless they are connected with a material that conducts. - Once the plates are connected, the capacitor will discharge.

Applications of Potential Difference

- One common application of the concept of potential difference is in the operation of electrical circuits. - Recall that the reference point for determining the electric potential at some point is arbitrary and must be defined. - Earth is frequently designated to have an electrical potential of zero and makes a convenient reference point. - Thus, grounding an electrical device (connecting it to Earth) creates a possible reference point, which is commonly used to measure the electric potential in an electric circuit.

Uniform Field

A field that has the same value and direction at all points

Capacitor

- A device that is used to store electrical potential energy. - An energized (or charged) capacitor is useful because energy can be reclaimed from the capacitor when needed for a specific application. - A typical design for a capacitor consists of two parallel metal plates separated by a small distance. This type of capacitor is called a *parallel-plate capacitor*. - When we speak of the *charge on a capacitor*, we mean the *magnitude of the charge on either plate* - The capacitor is energized by connecting the plates to the two terminals of a battery or other sources of potential difference. - When this connection is made, charges are removed from one of the plates, leaving the plate with a net charge. - An equal and opposite amount of charge accumulates on the other plate. - Charge transfer between the plates stops when the potential difference between the plates is equal to the potential difference between the terminals of the battery. - When connected to a battery, the plates of a parallel-plate capacitor become oppositely charged

Resistance depends on length, area, material, and temperature

- A longer length of wire provides more resistance than a shorter length of wire does. - A wider wire allows charges to flow more easily than a thinner wire does. - Resistance increases as the temperature of the metal increases. - Shorter length = less resistance - Greater length = more resistance - Wider wire = less resistance - Thinner wire = more resistance - Copper = less resistance - Iron = more resistance - Low temperature = lower resistance - Higher temperature = higher resistance

The material between a capacitor's plates can change its capacitance

- So far, we have assumed that the space between the parallel plates of a parallel-plate capacitor is a vacuum. However, in many parallel-plate capacitors, this space is filled with a material called a *dielectric*. - A dielectric is *an insulating material, such as air, rubber, glass, or waxed paper.* - When a dielectric is inserted between the plates of a capacitor, the capacitance increases. - The capacitance increases because the molecules in a dielectric can align with the applied electric field, causing an excess negative charge near the surface of the dielectric at the positive plate and an excess positive charge near the surface of the dielectric at the negative plate. - The surface charge on the dielectric effectively reduces the charge on the capacitors plates. Thus, the plates can store more charge for a given potential difference. - According to the expression Q = C∆V, if the charge increases and the potential difference is constant, the capacitance must increase. - *A capacitor with a dielectric can store more charge and energy for a given potential difference than can the same capacitor without a dielectric.*

Capacitance is the ratio of charge to potential difference

- The ability of a conductor to store energy in the form of electrically separated charges is measured by the *capacitance* of the conductor. - Capacitance is defined as the ratio of the net charge on each plate to the potential difference created by the separate charges.

Capacitance For a Parallel-Plate Capacitor in a Vacuum (Equation)

- The capacitance of a parallel-plate capacitor with no material between its plates is given by the following expression: capacitance = (permittivity of a vacuum) x (area of one of the plates / distance between the plates) - In this expression, the Greek letter ε (epsilon) represents a constant called the permittivity of the medium. When it is followed by a subscripted zero, it refers to a vacuum. It has a magnitude of 8.85 x 10⁻¹² C²/N∙m²

Potential Difference

- The concept of electrical potential energy is useful in solving problems, particularly those involving charged particles. But at any point in an electric field, as the magnitude of the charge increases, the magnitude of the associated electrical potential energy increases. - It is more convenient to express the potential in a manner independent of the charge at that point, a concept called *electric potential*. - The electric potential at some point is defined as the electrical potential energy associated with a charged particle in an electric field divided by the charge of the particle. V = PEelectric / q - The potential at a point is the result of the fields due to all other charges near enough and large enough to contribute force on a charge at that point. - In other words, the electrical potential at a point is independent of the charge at that point. - The force that a test charge at the point in question experiences is proportional to the magnitude of the charge.

The superposition principle can be used to calculate the electric potential for a group of charges

- The electric potential at a point near two or more charges is obtained by applying a rule called the *superposition principle* - This rule states that the total electric potential at some point near several point charges is the algebraic sum of the electric potentials resulting from each of the individual charges. - While this is similar to the method used previously to find the resultant electric field at a point in space, here the summation is easier to evaluate because the electric potentials are scalar quantities, not vector quantities. There are no vector components to consider. - To evaluate the electric potential at a point near a group of point charges, you simply take the algebraic sum of the potentials resulting from all charges. Remember, you must keep track of signs. - The electric potential at some point near a positive charge is positive, and the potential near a negative charge is negative.

The potential difference in a uniform field varies with the displacement from a reference point

- The expression for potential difference can be combined with the expression for electrical potential energy. - The resulting equations are often simpler to apply in certain situations. - Consider the electrical potential energy of a charge in a uniform electric field PEelectric = -qEd This expression can be substituted into the equation for potential difference ΔV = Δ(-qEd) / q - As the charge moves in an electric field, the quantity in the parentheses does not change from the reference point. Thus, the potential difference in this case can be rewritten as follows: ∆V = -Ed potential difference = -(magnitude of the electric field x displacement) - Keep in mind that d is the displacement parallel to the field and that motion perpendicular to the field does not change the electric potential energy

Drift speeds are relatively small

- The magnitudes of drift velocities are typically very small. In fact, the drift speed is much less than the average speed between collisions - For example, in a copper wire that has a current of 10.0 A, the drift speed of electrons is only 2.46 x 10⁻⁴ m/s. These electrons would take about 68 min to travel 1 m. The electric field, on the other hand, reaches electrons throughout the wire at a speed approximately equal to the speed of light.

Current is the rate of charge movement

- The movement of electric charge is known as current - A current exists whenever there is a net movement of electric charge through a medium - To define current more precisely, suppose electrons are moving through a wire. The *electric current* is the rate at which these charges move through the cross section of the wire. - If ∆Q is the amount of charge that passes through this area in a time interval, ∆t, then the current, I, is the ratio of the amount of charge to the time interval. - Note that the direction of the current is opposite the movement of the negative charges.

Conventional current is defined in terms of positive charge movement

- The moving charges that make up a current can be positive, negative, or a combination of the two. - In a common conductor, such as copper, current is due to the motion of negatively charged electrons, because the atomic structure of solid conductors allows the electrons to be transferred easily from one atom to the next. - In certain particle accelerators, a current exists when positively charged protons are set in motion. - In some cases--in gases and dissolved salts, for example--current is the result of positive charges moving in one direction and negative charges moving in the opposite direction. - Positive and negative charges in motion are sometimes called *charge carriers* - *Conventional current* is defined in terms of the flow of positive charges. Thus, negative charge carriers, such as electrons, would have a conventional current in the direction opposite their physical motion.

The reference point for potential difference near a point charge is often at infinity

- To determine the potential difference between two points in the field of a point charge, first calculate the electric potential associated with each point. - Imagine a point charge q2 at point A in the electric field of a point charge q1 at point B some distance, r, away - The electric potential at point A due to q1 can be expressed as follows: Va = PEelectric/q2 = kcq1q2/rq2 = kcq1/r - The charge q1 is responsible for the electric potential at point A. *Therefore, an electric potential exists at some point in an electric field regardless of whether there is a charge at that point.* - In this case, the electric potential at a point depends on only two quantities: the charge responsible for the electric potential (in this case q1) and the distance r from this charge to the point in question

Drift velocity is the net velocity of charge carriers

- To see how electrons move, consider a solid conductor in which the charge carriers are free electrons. - When the conductor is in electrostatic equilibrium, the electrons move randomly, similar to the movement of molecules in a gas. - When a potential difference is applied across the conductor, an electric field is set up inside the conductor. The force due to that field sets the electrons in motion, thereby creating a current. - These electrons do not move in straight lines along the conductor in a direction opposite the electric field. Instead, they undergo repeated collisions with the vibrating metal atoms of the conductor.

Capacitance depends on the size and shape of the capacitor

- We can combine the two equations for capacitance to find an expression for the charge stored on a parallel-plate capacitor. Q = (ε₀A/d)ΔV - This equation tell us that for a given potential difference, ΔV, the charge on a plate is proportional to the area of the plates and inversely proportional to the separation of the plates. - Suppose an isolated conducting sphere has a radius R and a charge Q. The potential difference between the surface of the sphere and infinity is the same as it would be for an equal point charge at the center of the sphere. ∆V = kc(Q/R) - Substituting this expression into the defintion of capacitance results in the following expression: Csphere = Q/∆V = R/kc - This equation indicates that the capacitance of a sphere increases as the size of the sphere increases. - Because the Earth is so large, it has an extremely large capacitance. Thus, Earth can provide or accept a large amount of charge without its electric potential changing too much. This is the reason why Earth is often used as a reference point for measuring potential differences in electric circuits.

Resistance to Current

- When a lightbulb is connected to a battery, the current in the bulb depends on the potential difference across the battery. - For example, a 9.0V battery connected to a light bulb generates a greater current than a 6.0 V battery connected to the same bulb. - The materials that make up the connecting wires and the bulb's filament also affect the current in the bulb. Even though most materials can be classified as conductors or insulators, some conductors allow charges to move through them more easily than others. - The opposition to the motion of charges through a conductor is the conductor's resistance.

Electrical potential energy is similar to gravitational potential energy

- When electrical potential energy is calculated, d is the magnitude of the displacement's component in the direction of the electric field. - The electric field does work on a positive charge by moving the charge in the direction of E (just as Earth's gravitational does work on a mass by moving the mass toward Earth). - After such a movement, the system's final potential energy is less than its initial potential energy - A negative charge behaves in the opposite manner, because a negative charge undergoes a force in the opposite direction - Moving a charge in a direction that is perpendicular to E is analogous to moving an object horizontally in a gravitational field: no work is done, and the potential energy of the system remains constant

Electrolyte

A solute that dissolves in water to give a solution that conducts electric current

Unit for electrical potential energy

Joule (J) For Work: J = N∙m J = kg∙m²/s²

Electrical Potential Energy in a Uniform Electric Field (Equation)

PEelectric = -qEd electrical potential energy = -(charge x electric field strength x displacement from the reference point in the direction of the field) Units: Joules (J) q = charge (C) E = electric field strength (N/C) d = distance (m)

Potential Difference

The work that must be performed against electric forces to move a charge between the two points in question, divided by the charge.

Electric Potential

The work that must be performed against electric forces to move a charge from a reference point to the point in question, divided by the charge

Capacitance (Equation)

capacitance = (magnitude of the charge on each plate) / (potential difference) The SI unit for capacitance is the farad, F, which is equivalent to a coulomb per volt (C/V). In practice, most typical capacitors have capacitances ranging from microfarads (1µF = 1 x 10⁻⁶ F) to picofarads (1 pF = 1 x 10⁻¹² F)

Electric Current (Equation)

electric current = charge passing through a given area / time interval - The SI unit for current is the ampere, A - One ampere is equivalent to one coulomb of charge passing through a cross-sectional area in a time interval of one second 1 A = 1C/s

Electrical Potential Energy Stored in a Charged Capacitor (Equation)

electrical potential energy = ½(charge on one plate)(final potential difference) - Note that this equation is also an expression for the work required to charge the capacitor. - By substituting the definition of capacitance (C = Q/ΔV), we can see that these alternative forms are also valid: PEelectric = ½C(ΔV)² PEelectric = Q²/2C

Resistance (Equation)

resistance = potential difference / current - The SI unit for resistance, the ohm, is equal to one volt per ampere and is represented by the Greek letter omega