Chapter 23: Electric Potential Energy
What kind of integral represents the work done in moving a particle from point a to point b?
A line integral.
What are two ways of interpreting electric potential energy?
1. The potential energy difference Ua - Ub equals the work that is done by the electric force when the particle moves from a to b. 2. The potential energy difference is defined as the work that must be done by an external force to move the particle slowly from b to a against the electric force.
What is the relationship between the electric field and an equipotential surface?
Because potential energy does not change as a test charge moves over an equipotential surface, the electric field can do no work on such a charge. Since it does not work, E must be perpendicular to the surface at every point so that the electric force q0E is always perpendicular to the displacement of a charge moving on the surface.
What is the relationship between the strength of the field and the distance between equipotential surfaces?
In regions where the field is weaker, the equipotential surfaces are farther apart, and vice versa. This is directly analogous to the downhill force of gravity being greatest in regions on a topographic map where contour lines are close together.
What does the work done during a small displacement dl depend on?
It depends only on the change dr in the distance r between the charges, which is the radial component of the displacement.
What is an equipotential surface?
It is a three-dimensional surface on which the electric potential V is the same at every point. That is, if a test charge q0 is moved from point to point on such a surface, the electric potential energy q0 V remains constant.
What does the work-energy theorem say?
It says that the change in kinetic energy dK = Kb-Ka during a displacement equals the total work done on the particle.
What is the relationship between the direction of an electric field and V?
Moving with the direction of E means moving in the direction of decreasing V, and moving against the direction of E means moving in the direction of increasing V.
Does is it make sense to say, "The electric potential energy of a point charge"?
No, because the potential energy is a shared property of the two charges. If the distance q and q0 is changed from ra to rb, the change in potential energy is the same whether q is held fixed and q0 is moved or q0 is head fixed and q is moved.
Does there have to be a charge at a give point for a potential V to exist at that point?
No, in the same way an electric field can exist at a given point even if there's no charge there to respond to it.
What is potential?
Potential is the potential energy per unit charge. V = U/q0 Potential is a scalar.
What do we mean when we say that when all charges are at rest, the entire solid volume of a conductor is at the same potential?
Since E = 0 everywhere inside the conductor, the integral is guaranteed to be zero for any two such points a and b inside the conductor. Hence the potential is the same for any two points within the solid volume of the conductor.
What is the work done by a conservative force?
W = -dU When W is positive, Ua is greater than Ub, dU is negative, and the potential decreases. Work done by a conservative force is independent of whatever path it takes.
What is work the work on a particle in a uniform field?
W = Fd = q0Ed U = q0 = Ey
Why is it that when all charges are at rest, the surface of a conductor is always an equipotential surface?
We can prove this statement by showing that when all charges are at rest, the electric field just outside a conductor must be perpendicular to the surface at every point. We know that E = 0 everywhere inside the conductor; otherwise, charges would move. In particular, at any point just inside the surface the component of E tangent to the surface is zero. It follows that the charge could move around a rectangular path partly inside and partly outside and return to its starting point with a net amount of work having been done on it. This would violate the conservative nature of electrostatic fields, so the tangential component of E just outside the surface must be zero at every point on the surface. Thus E is perpendicular to the surface at each point.
Why is it that for every electric field, the force exerted by that field is conservative?
We can represent any charge distribution as a collection of point charges, so we can always find a potential-energy function for any static electric field.
Why do we use the potential gradient?
We use the potential gradient when we know the electric potential and want to find the electric field. It basically states that If the potential V is known as a function of coordinates x, y, and z, the components of electric field E at any point are given by partial derivatives of V.
When is work positive, negative, or zero?
Work is positive when the force is in the same direction as the displacement. Work is negative when the force is in the opposite direction of the displacement. Work is zero when the force is perpendicular to the displacement.
When is U = 0 between two point charges?
U is zero when q and q0 are infinitely far apart and r = infinity.
What is the voltage?
Vab, the potential of a with respect to be, equals the work done by the electric force when a UNIT charge moves from a to b. In other words, Vab, the potential of a with respect to b, equals the work that must be done to move a UNIT charge slowly from b to a against the electric force.