Relationship between Electric Field and Potential, Conservative Fields and Uniform Electric Fields Worksheet

 

Electric Field & Electric Potential Worksheet

Topics Covered: Relationship between Electric Field and Potential, Conservative Fields, Uniform Electric Field

Part A: Theory Questions

Q1. Define electric potential difference in terms of work done.

Q2. Write the relation between electric field and potential.

Q3. Why is electric field equal to negative gradient of potential?

Q4. What does the negative sign in \( \vec{E} = -\nabla V \) indicate?

Q5. Define a conservative electric field.

Q6. State two properties of a conservative field.

Q7. Work done in moving a charge in a closed loop?

Q8. Can electric field lines intersect?

Q9. Relation between equipotential surface and electric field?

Q10. Electric field and potential inside a conductor?

Part B: Numerical Questions

Q11. \( V = 5x^2 \). Find electric field.

Q12. \( V = 3x + 4y \). Find electric field.

Q13. Potential difference 20 V across 5 m. Find electric field.

Q14. Electron moves through 100 V. Work done?

Q15. Field = \( 200 \, N/C \). Distance = 2 m. Find potential difference.

Q16. \( V = \frac{10}{x} \). Find electric field at \( x = 2 \).

Q17. Charge 2 C, potential difference 15 V. Work done?

Q18. \( E = 4x \). Find potential function.

Q19. Potential difference = 0. Work done for 5 C?

Q20. \( E = 50 \, N/C \), distance 3 m, charge 1 C. Work done?

Answer Key

Theory:

1. Work per unit charge
2. \( \vec{E} = -\nabla V \)
3. Rate of decrease of potential
4. Field from high to low potential
5. Path independent work
6. Path independence, zero work in loop
7. Zero
8. No
9. Perpendicular
10. \( E = 0 \), constant potential

Numerical:

11. \( -10x \)
12. \( -3\hat{i} - 4\hat{j} \)
13. 4 N/C
14. \( 1.6 \times 10^{-17} \, J \)
15. 400 V
16. 2.5 N/C
17. 30 J
18. \( -2x^2 + C \)
19. 0
20. 150 J

Detailed Solutions

Q11
\( E = -\frac{dV}{dx} \)
\( E = -\frac{d}{dx}(5x^2) = -10x \)

Q12
\( E_x = -\frac{\partial V}{\partial x} = -3 \)
\( E_y = -\frac{\partial V}{\partial y} = -4 \)

Q13
\( E = \frac{V}{d} = \frac{20}{5} = 4 \, N/C \)

Q14
\( W = qV = (1.6 \times 10^{-19})(100) = 1.6 \times 10^{-17} \, J \)

Q15
\( V = Ed = 200 \times 2 = 400 \, V \)

Q16
\( E = -\frac{d}{dx}(10/x) = 10/x^2 \)
At \( x = 2 \): \( E = 2.5 \, N/C \)

Q17
\( W = qV = 2 \times 15 = 30 \, J \)

Q18
\( V = -\int 4x \, dx = -2x^2 + C \)

Q19
\( W = q\Delta V = 0 \)

Q20
\( W = qEd = 1 \times 50 \times 3 = 150 \, J \)