# AP Physics C Electricity : Electricity

## Example Questions

### Example Question #7 : Calculating Electric Potential

Three identical point charges with  are placed so that they form an equilateral triangle as shown in the figure. Find the electric potential at the center point (black dot) of that equilateral triangle, where this point is at a equal distance, , away from the three charges.

Explanation:

The electric potential from point charges is .

Knowing that all three charges are identical, and knowing that the center point at which we are calculating the electric potential is equal distance from the charges, we can multiply the electric potential equation by three.

Plug in the given values and solve for .

### Example Question #8 : Calculating Electric Potential

Eight point charges of equal magnitude  are located at the vertices of a cube of side length . Calculate the potential at the center of the cube.

Explanation:

By the Pythagorean theorem, each charge is a distance

from the center of the cube, so the potential is

### Example Question #9 : Calculating Electric Potential

An infinite plane has a nonuniform charge density given by . Calculate the potential at a distance  above the origin.

You may wish to use the integral:

Explanation:

Use polar coordinates with the given surface charge density, and area element . Noting that a point  from the origin is a distance  from the point of interest, we calculate the potential as follows, integrating with respect to  from  to .

(by limit)

Remark: This is exactly the charge distribution that would be induced on an infinite sheet of (grounded) metal if a negative charge  were held a distance  above it.

### Example Question #10 : Calculating Electric Potential

A nonuniformly charged hemispherical shell of radius  (shown above) has surface charge density . Calculate the potential at the center of the opening of the hemisphere (the origin).

Explanation:

Use spherical coordinates with the given surface charge density , and area element . Every point on the hemispherical shell is a distance  from the origin, so we calculate the potential as follows, noting the limits of integration for  range from  to .

### Example Question #21 : Electricity

Three equal point charges  are placed at the vertices of an equilateral triangle of side length . Calculate the potential at the center of the triangle, labeled P.

Explanation:

Draw a line from the center perpendicular to any side of the triangle. This line divides the side into two equal pieces of length . From the center, draw another line to one of the vertices at the end of this side. This produces a 30-60-90 triangle with longer leg , so the hypotenuse (the distance from the vertex to the center) is . The potential at the center is due to three of these charges, so it must be

### Example Question #22 : Electricity

A uniformly charged hollow disk has inner radius  and outer radius , and carries a total charge . Calculate the potential a distance  from the center, on the axis of the disk.

Explanation:

Use a polar coordinate system with surface charge density  and area element . The distance from the point of interest to a point a distance  from the center is , so the potential is

### Example Question #23 : Electricity

Two point charges  and  are separated by a distance . Calculate the potential at point P, a distance  from charge  in the direction perpendicular to the line connecting the two charges.

Explanation:

By the Pythagorean theorem, the distance from point P to charge  is . Because point P is also  from charge , it follows that the potential is

### Example Question #24 : Electricity

A nonuniformly charged ring of radius  carries a linear charge density of . Calculate the potential at the center of the ring.

Explanation:

Use a polar coordinate system, the given linear charge density , and length element . Since every point on the ring is the same distance  from the center, we calculate the potential as

### Example Question #25 : Electricity

In this model of a dipole, two charges  and  are separated by a distance  as shown in the figure, where the charges lie on the x-axis at  and  respectively. Calculate the exact potential a distance  from the origin at angle  from the axis of the dipole.

Explanation:

By the law of cosines, the distance from the point to charge  is

.

The distance to charge  can be found by using the law of cosines using the supplementary angle , for which . Therefore the distance to  is

.

Lastly, the exact potential is given by

.

Remark: Far from the dipole (approximating ) gives the much simpler equation for the potential of an ideal electric dipole .

### Example Question #26 : Electricity

A uniformly charged hollow spherical shell of radius  carries a total charge . Calculate the potential a distance  (where ) from the center of the sphere.