# AP Physics C Electricity : Calculating Electric Potential

## Example Questions

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### Example Question #1 : Calculating Electric Potential

A proton moves in a straight line for a distance of . Along this path, the electric field is uniform with a value of . Find the potential difference created by the movement.

The charge of a proton is .

Possible Answers:

Correct answer:

Explanation:

Potential difference is given by the change in voltage

Work done by an electric field is equal to the product of the electric force and the distance travelled. Electric force is equal to the product of the charge and the electric field strength.

The charges cancel, and we are able to solve for the potential difference.

### Example Question #1 : Calculating Electric Potential

For a ring of charge with radius  and total charge , the potential is given by .

Find the expression for electric field produced by the ring.

Possible Answers:

Correct answer:

Explanation:

We know that .

Using the given formula, we can find the electric potential expression for the ring.

Take the derivative and simplify.

### Example Question #1 : Calculating Electric Potential

The potential outside of a charged conducting cylinder with radius  and charge per unit length  is given by the below equation.

What is the electric field at a point located at a distance from the surface of the cylinder?

Possible Answers:

Correct answer:

Explanation:

The radial electric field outside the cylinder can be found using the equation .

Using the formula given in the question, we can expand this equation.

Now, we can take the derivative and simplify.

### Example Question #1 : Calculating Electric Potential

A proton moves in a straight line for a distance of . Along this path, the electric field is uniform with a value of . Find the work done on the proton by the electric field.

The charge of a proton is .

Possible Answers:

Correct answer:

Explanation:

Work done by an electric field is given by the product of the charge of the particle, the electric field strength, and the distance travelled.

We are given the charge (), the distance (), and the field strength (), allowing us to calculate the work.

### Example Question #1 : Calculating Electric Potential

A negative charge of magnitude  is placed in a uniform electric field of , directed upwards. If the charge is moved  upwards, how much work is done on the charge by the electric field in this process?

Possible Answers:

Correct answer:

Explanation:

Relevant equations:

Given:

First, find the potential difference between the initial and final positions:

2. Plug this potential difference into the work equation to solve for W:

### Example Question #1 : Calculating Electric Potential

Three point charges are arranged around the origin, as shown.

Calculate the total electric potential at the origin due to the three point charges.

Possible Answers:

Correct answer:

Explanation:

Electric potential is a scalar quantity given by the equation:

To find the total potential at the origin due to the three charges, add the potentials of each charge.

### Example Question #1 : 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.

Possible Answers:

Correct answer:

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 #1 : 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.

Possible Answers:

Correct answer:

Explanation:

By the Pythagorean theorem, each charge is a distance

from the center of the cube, so the potential is

### Example Question #1 : 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:

Possible Answers:

Correct answer:

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 #1 : 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).

Possible Answers:

Correct answer:

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 .

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