Q1. What is the electric potential at a distance above the center of a disk of uniform surface charge density and radius ?

Background

Topic: Electrostatics – Electric Potential of Continuous Charge Distributions

This question tests your ability to calculate the electric potential due to a continuous distribution of charge (a disk), using integration and the principle of superposition.

Key Terms and Formulas:

• Surface charge density (): Charge per unit area on the disk.

• Electric potential (): The work done per unit charge to bring a test charge from infinity to a point in space.

• Superposition principle: The total potential is the sum of potentials from each infinitesimal charge element.

• Potential from a point charge: , where .

• For a continuous distribution: .

Step-by-Step Guidance

1. Draw a diagram showing the disk, its radius , surface charge density , and the point at distance above the center. Visualize the disk as composed of concentric rings.

2. Consider a ring of radius and thickness on the disk. The area of this ring is , so the charge on the ring is .

3. The distance from a point on the ring to the point above the center is .

4. Write the expression for the potential due to this ring: .

5. Set up the integral for the total potential: .

6. At this point, you should solve the integral to find the explicit expression for . Try to carry out the integration and simplify the result.

Try solving on your own before revealing the answer!

Q2. What is the electric field of the disk at the distance above the center of a disk of uniform surface charge density and radius ? (Find both magnitude and direction.)

Background

Topic: Electrostatics – Electric Field from Continuous Charge Distributions

This question tests your ability to find the electric field from a continuous charge distribution, using the relationship between electric potential and electric field.

Key Terms and Formulas:

• Electric field (): The force per unit charge at a point in space.

• Relationship to potential: (for the -component above the disk).

• Superposition principle: The total field is the sum of fields from each infinitesimal charge element.

Step-by-Step Guidance

1. Recall the expression for the potential you found in the previous question.

2. To find the electric field in the direction, use .

3. Differentiate the potential with respect to : .

4. Carefully compute the derivative, paying attention to the chain rule for .

5. After differentiating, simplify the expression to find the magnitude and direction of the electric field at distance above the disk.

Try solving on your own before revealing the answer!

Q3. Four charges lie on the corners of a square with sides of length . What is the electric potential at the center of the square?

Background

Topic: Electrostatics – Superposition Principle for Point Charges

This question tests your ability to use the superposition principle to find the potential at a point due to multiple point charges.

Key Terms and Formulas:

• Electric potential from a point charge: .

• Superposition principle: Add the potentials from each charge at the center.

• Distance from each corner to the center: .

Step-by-Step Guidance

1. Draw a diagram of the square, label the charges at each corner, and mark the center.

2. Calculate the distance from each corner to the center: .

3. Write the expression for the potential at the center due to one charge: .

4. Use the superposition principle: Add the potentials from all four charges.

5. Set up the sum: .

6. Substitute the value of and simplify the expression. Complete the calculation to find the potential at the center.

Try solving on your own before revealing the answer!