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Electric Potential Energy

- When a force acts on a particle that moves from point a to point b, the work Wab done by the force is given by a line integral:
- dx is a very small distance along the particle’s path.
- angle is the angle between F and dx at each point along the path.
- If the force F is conservative, the work done by F can be expressed in terms of potential energy U.
- When the particle moves from a point where the potential energy is Ua to a point where it is Ub, the change in potential energy is DU = Ub – Ua.

The work done by the force is:

- When Wab is positive, Ua is greater than Ub, DU is negative, and the potential energy decreases.
- Work is positive when the force and the direction of motion are in the same direction (parallel; = º); work is negative when the force and the direction of motion are in opposite directions (antiparallel; = 180º).
- The work-energy theorem says that the change in kinetic energy K = Kb – Ka during any displacement is equal to the total work done on the particle.

If the only work done on the particle is done by conservative forces, then Kb – Ka = -(Ub – Ua), which can be rewritten as Ka + Ua = Kb + Ub.

- The total mechanical energy is conserved.

Electric Potential Energy in a Uniform Electric Field

- A pair of charged parallel metal plates sets up a uniform, downward electric field with magnitude E.

The field exerts a downward force F = qo·E on a positive test charge qo.

- As the charge moves downward a distance d from point a to b, the force on the test charge is constant and independent of its location.
- Work done by the electric field is Wab = F·d = qo·E·d.
- Work is positive because the force is in the same direction as the displacement.

The electric force on the charge is constant, therefore, the force exerted on qo is conservative.

- The work done by the electric field is independent of the path the particle takes from point a to point b.
- For a positive charge in a uniform electric field:
- If the positive charge moves in the direction of the electric field E, the electric field does positive work on the charge and the potential energy U decreases; K of the charge increases.
- If the positive charge moves in the direction opposite the electric field E, the electric field does negative work on the charge and the potential energy U increases.

If a negative charge moves in the direction of the electric field E, the electric field does negative work on the charge and the potential energy U increases.

- If a negative charge moves in the direction opposite to the direction of the electric field E, the electric field does positive work on the charge and the potential energy U decreases; K of the particle increases.
- Whether the test charge is positive or negative:
- U increases if the test charge qo moves in the direction opposite to the electric force F = qo·E.
- U decreases if the test charge qo moves in the same direction as F = qo·E.

Electric Potential Energy of Two Point Charges

- To calculate the work done on a test charge qo moving in the electric field caused by a single stationary point charge q.
- For a displacement along the radial line from point a to point b, the force on qo is given by Coulomb’s law:

If q and qo have the same sign, the force F is repulsive and F is positive; if q and qo have opposite signs, the force F is attractive and F is negative.

- The force F is not constant during the displacement from point a to point b, so we have to integrate to determine Wab:

The work done by the electric force for a particular path depends only on the end points.

The work is the same for all possible paths from a to b.

- Consider the displacement in which a and b do not lie on the same radial line:

Work done on qo during the displacement given by:

- The work done during a small displacement dl depends only on the change dr in the distance r between the charges, which is the radial component of the displacement.
- The work done on qo by the electric field E produced by q depends only on the displacement between ra and rb, not on the path from ra to rb.

If qo returns to its starting point a by any path, the total work done is zero (displacement is zero); integral from ra to ra = .

- The potential energy Ua when qo is at any distance r from q is:
- The potential energy is positive when q and qo have the same sign and negative if they have opposite signs.
- Potential energy is always defined relative to some reference point where U = .
- U = when q and qo are infinitely far apart (r = ∞).
- U represents the work that would be done on the test charge qo by the electric field of q if qo moved from an initial distance r to infinity.

Left figure: If q and qo have the same sign, the force is repulsive and the work done by the field of q is positive and U is positive at any finite separation.

- Right figure: If q and qo have opposite signs, the force is attractive and the work done by the field of q is negative and U is negative at any finite separation.

Electric Potential Energy with Several Point Charges

- If qo moves in an electric field E produced by point charges q1, q2, and q3 at distances r1, r2, and r3 from qo:

the total electric field at

each point is the vector

sum of the fields due to

the individual charges, and

the total work done on qo

during any displacement is

the sum of the contributions

of the individual charges.

Work and potential energy are scalar quantities, not vector quantities; keep the negative signs!

- The potential energy associated with test charge qo at point a is the algebraic sum:
- Two viewpoints on electric potential energy:
- In terms of the work done by the electric field on a charged particle moving in the field, when a particle moves from point a to point b, the work done on it by the field is Wab= Ua – Ub.
- The potential energy difference Ua – Ub is equal to the work that is done by the electric force when the particle moves from point a to point b.

When Ua is greater than Ub, the field does positive work on the particle as it “falls” from a point of higher potential energy a to a point of lower potential energy b.

- Alternative: consider how much work we would have to do to “raise” a particle from a point b where the potential energy is Ub to a point a where it has greater potential energy Ua (ex. Pushing 2 protons closer together).
- To move the particle slowly so as not to give it any kinetic energy, we need to exert an additional external force that is equal to and opposite in direction to the electric field force and does positive work.
- The potential energy difference Ua – Ub is then defined as the work that must be done by an external force to move the particle slowly from point b to point a against the electric force.

Because the external force is the negative of the electric field force and the displacement is in the opposite direction, the potential difference Ua – Ub is positive.

- This also works if Ua is less than Ub, corresponding to “lowering” the particle (ex. moving 2 protons away from each other); the potential difference Ua – Ub is negative.

Electric Potential

- Electric potential is potential energy per unit charge (J/C).
- Electric potential V at any point in an electric field is the potential energy U per unit charge associated with a test charge qo at that point:

Potential energy and charge are both scalar quantities, so electric potential is a scalar too..

- SI unit of electric potential is the volt; 1 V = 1 J/C
- Dividing Wab by qo:
- Va is the potential at point a and Vb is the potential at point b.
- The work done per unit charge by the electric force when a charge moves from a to b is equal to the potential at a minus the potential at b.

The difference Va – Vb is called the potential of a with respect to b; sometimes abbreviated as Vab = Va – Vb and called the potential difference between a and b.

- In electric circuits, the potential difference between 2 points is called voltage.
- Vab, the potential of a wrt b, equals the work done by the electric force when a unit charge (1 C) moves from a to b.

Alternative: Vab, potential of a with respect to b, equals the work that must be done to move a unit charge (1 C) slowly from b to a against the electric force.

- Instrument used the measure the potential difference (voltage) between two points is called a voltmeter

What Good is the Electric Potential?

Two essential ideas:

- Electric potential depends only on the source charges and their geometry. Potential is the “ability” of the source charges to have an interaction if a charge q shows up. This ability, or potential, is present throughout space regardless of whether or not charge q is there to experience it.

If we know the electric potential V throughout a region of space, we know the potential energy U = q·V of any charge that enters the region of space.

- Charged particles speed up or slow down as they move through a region of changing potential.
- Conservation of energy in terms of electric potential:

Ki + q·Vi = Kf + q·Vf

Calculating the Potential from the Field

- The potential difference between any initial point i and final point f can be determined if we know the electric field vector E along any path connecting i and f.
- Find the work done on a positive test charge by the electric field as the charge moves from i to f.

For the electric field represented by the field lines in the figure, a positive test charge qo moves along the path shown from point i to point f.

At any point on the path, a force qo·E acts on the charge as it moves through a differential displacement ds.

- The differential work dW done on the charge by the force during the displacement ds is:

dW = F•ds = qo·E·ds

- The total work W done on the particle by the electric field as the particle moves from point I to point f is the sum of the differential works done on the charge as it moves through all the displacements ds along the path:

- Electric field lines point from regions of high potential to regions of low potential, thus the negative sign.

The potential difference Vf – Vi between any two points i and f in an electric field is equal to the negative of the line integral of E•ds from i to f.

- Because the electrostatic force is conservative, all paths produce the same result.
- Allows us to determine the

difference in potential between any two points in the electric field.

- If we set Vi = , then gives us the potential V at any point f in the electric field relative to the zero potential at point i.

The negative sign indicates that point b is at a lower potential than point a; that is, Vb < Va.

- As a test charge qo moves from a to b, the change in the potential energy is:
- If qo is positive, DU is negative.
- For a positive charge in an electric field, the charge will lose electric potential energy when it moves in the direction of the electric field as it gains kinetic energy due to the electric force qo·E it experiences in the direction of E.

The loss of electric potential energy is equal to the gain in kinetic energy.

If the test charge is negative, then DU is positive. The test charge gains electric potential energy when it moves in the direction of the electric field.

- If a negative charge is released in the electric field, it accelerates in the direction opposite the electric field.

The Accelerations of Positive and Negative Charges

- A positive charge will accelerate from a region of higher potential toward a region of lower potential.
- A negative charge will accelerate from a region of lower potential to a region of higher potential.
- For example: a 12 V car battery connected to the headlights. The positive terminal has a potential that is 12 V higher than the potential at the negative terminal. Positive charges are repelled from the positive terminal and travel through the wires and headlight toward the negative terminal.

As the charges pass through the headlight, almost all their potential energy is converted into heat, which causes the filament to glow “white hot” and emit light. When the charges reach the negative terminal, they no longer have any potential energy.

- The battery then provides the charges with additional potential energy by moving them to the higher potential terminal, and the cycle is repeated. In raising the potential energy of the charges, the battery does work WAB on them, and draws from its reserve of chemical energy to do so.

Historically, it was believed that positive charges flow in the wires of an electric circuit. We now know that negative charges flow in wires from the negative terminal toward the positive terminal. It is very common, particularly when referring to the direction of current flow using Kirchhoff’s laws, to describe the flow of negative charges by specifying the opposite, but equal, flow of positive charges. The hypothetical flow of positive charges is called the conventional electric current.

Electric Potential of a Point Charge

- The electric potential due to a point charge Q is given by:
- If Q is positive, the potential is positive; if Q is negative, the potential is negative.
- Voltage is a scalar quantity, therefore, only the magnitude of the potential difference is important, not the direction.

Equipotential Surface

- Equipotential surfaces are surfaces on which the potential is everywhere the same. No work is done in moving a charge over an equipotential surface. The electric field at an equipotential surface must be perpendicular to the surface since otherwise there would be a component of the field and also therefore an electric force parallel to the equipotential surface. Then work would be done in moving charges over the surface and the surface would therefore not be an equipotential surface.

Equipotential Surface: A circle of radius "r" around a charge where the electric field strength is the same, therefore, the potential is the same. An equipotential surface consists of a continuous distribution of points having the same potential. On the diagram, the equipotential lines are the dotted lines around the charges.Equipotential Surface

Work is done by or against the field only when a charge is moved from one equipotential point to another. This is analogous to moving one from level of gravitational potential energy to another. Equipotential Surface

When moving a charge along an equipotential line, DV = 0 V, therefore, DU = qo·DV = qo·0 V.Equipotential Surface

A unit of energy commonly used in atomic and nuclear physics is the electron-volt. An electron-volt (eV) is the energy that an electron (or proton) gains when moving through a potential difference of 1 V.

1 eV = 1.602 x 10-19 J

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