Relation between E and V - UNSOLVED PRACTICE SET

Q7. Write the relation between electric field and potential in differential form. Explain the physical meaning of the negative sign.

Q8. If the electric potential in a region is given by V = 4x + 2y volts, find the electric field vector in that region.

Q9. The electric field is the negative gradient of potential. What does "gradient" mean in this context? Explain in simple terms.

Q10. In a certain region, the potential is constant (V = 10 V everywhere). What is the electric field in this region? Explain.

Q11. A student measures the potential difference between two points 5 cm apart in a uniform electric field and finds it to be 25 V. Calculate the electric field.

Q12. The potential in a region varies as V = 10/r volts, where r is in meters. Find the electric field at r = 2 m.

Q13. Derive the relation E = −dV/dr for a uniform electric field. Explain why the electric field is the negative gradient of potential and what this means physically.

Q14. The electric potential in a region is given by V = 3x²y − 2yz³ volts. Find the expression for the electric field vector E at any point (x, y, z). Calculate the electric field at the point (1, 2, 1) m.

Q15. In a school experiment, two parallel plates are connected to a battery maintaining a potential difference of 50 V. The plates are 5 cm apart.

(a) Calculate the uniform electric field between the plates.

(b) Write the expression for potential V as a function of distance x from the positive plate.

(c) If a proton is placed at the positive plate, calculate its acceleration.

(d) How does the potential vary if the plate separation is doubled while keeping the battery connected?

[Given: e = 1.6 × 10⁻¹⁹ C, mₚ = 1.67 × 10⁻²⁷ kg]

Q16. The electric potential in a certain region of space is given by V(x, y, z) = 2x² + 3y − 4z² volts.

(a) Find the components of the electric field Eₓ, Eᵧ, and Eᵤ.

(b) Calculate the magnitude of the electric field at the point (1, 1, 1) m.

(c) Find the direction of the electric field at this point (express as a unit vector).

(d) At what points in the x-y plane (z = 0) is the electric field purely in the z-direction?

Q17. In a physics lab, a student sets up an experiment where the electric potential along the x-axis is measured as V(x) = 100 − 20x + x² volts, where x is in meters.

(a) Find the expression for the electric field E(x).

(b) At what value of x is the electric field zero?

(c) Calculate the electric field at x = 0, x = 5 m, and x = 10 m.

(d) Plot a rough graph of E versus x for 0 ≤ x ≤ 15 m and explain its physical significance.

Q18. A spherical charge distribution has electric potential V(r) given by:

V(r) = (kQ/2R)(3 − r²/R²) for r ≤ R (inside)

V(r) = kQ/r for r ≥ R (outside)

where R is the radius of the sphere and Q is the total charge.

(a) Find the electric field for r ≤ R and r ≥ R.

(b) Show that the electric field is continuous at r = R.

(c) Calculate the electric field at r = R/2 and r = 2R.

(d) Sketch the graphs of V(r) and E(r) versus r.