Physics 1102 In-Class Problems: Electrostatic Potential

1. (a) E = 5x²i. Calculate ΔV between x = 0 m and x = 4 m.

   
   (b) E = (2/y)j. Calculate ΔV between y = 2 m and y = 5 m.

2. Find the electrostatic potential difference between points A and B which are distances r_A = 2.0 m and r_B = 1.0 m from an infinitely long thin wire with λ = 1.0 μC/m. The result E = λ/2pe₀r is useful. If an electron (q = -e = -1.602 × 10^-19 C and mass m_e = 9.11 × 10^-31 kg) is released from rest at point A, what is it's speed at point B?
3. Find the electrostatic potential between points A and B which are distances r_A = 2.0 m and r_B = 1.0 m from an infinitely large thin plate with Σ = 1.0 μC/m². The result E = Σ/2ε₀ is useful. If an electron (q = -e = -1.602 × 10^-19 C and mass m_e = 9.11 × 10^-31 kg) is released from rest at point A, what is it's speed at point B?.
4. In Gauss’ Law question #7, we found the electric field due to a uniformly charge long thick plate to be



where d was the thickness of the plate, ρ ₀ was the volume charge density, and a is the distance from the centre of the plate. Determine V(x) relative to V(0) where x is the distance from the centre of the plate. Assume V(0) = 0. Plot V as a function of x.

5. In Gauss’ Law question #8, we found the electric field due to a long thin wire of uniform charge density λ to be E(a) = (λ/2pe₀a)r where a is the radial distance from the wire and r is the unit radial vector. Determine V(r) relative to V(r=1) where r is the distance from the centre of the cylinder. Assume V(1) = 0. Plot V as a function of r.
6. In Gauss’ Law question #9, we found the electric field due to a long thin cylindrical shell of radius R and negative surface charge density Σ to be



where the field is radial (i.e. in the r direction). Determine V(r) relative to V(r = 0) where r is the distance from the centre of the cylinder. Assume V(0) = 0. Plot V as a function of r.

7. In Gauss’ Law question #10, we found the electric field due to a solid sphere of radius R and total charge Q to be radial and have the form



where the field is radial (i.e. in the r direction). Determine V(r) relative to V(¥) where r is the distance from the centre of the sphere. Assume V(r=¥) = 0. Plot V as a function of r.

8. In Gauss’ Law question #11, we found the electric field of a thick spherical shell of total charge Q and inner and outer radii of R and 2R to be



where the field is radial (i.e. in the r direction). Determine V(r) relative to V(¥)where r is the distance from the centre of the sphere. Assume V(¥) = 0. Plot V as a function of r.

9. In Gauss’ Law question #12, we found the electric field due to a thin spherical shell of radius R and surface charge -s to be



where the field is radial (i.e. in the r direction). Determine V(r) relative to V(r=¥) where r is the distance from the centre of the sphere. Assume V(r=¥) = 0. Plot V as a function of r.

10. The electric field due to a large plate is E = Σ/2ε₀ as we have seen. A capacitor consists of two identical plates with equal but opposite charge distributions.

(a) Show that the net electric field is zero outside the plates and Enet = Σ/ε₀ between the plates.
(b) Show that the potential difference between the plates is directly proportional to the separation d of the plates.
11. Two point charges of q₁ = 3.0 μC and q₂ = 4.0 μC are situated at the opposite corners of a rectangle as shown below. The short side has length L = 0.25 m. Find the total potential at points A and B. If a free particle of charge q_f = 1.0 μC and mass M = 15 g has speed v_A = 2.50 m/s at point A and it follows the indicated path, what will be its speed at point B?




12. Five charges are arranged as shown below. What is the electrostatic potential energy of each configuration? The separation between charges is L.


(a) 

(b) 

13. A rod is bent into a semi-circular arc of radius R. The rod has a uniform linear charge distribution λ. Find the potential at the centre of the arc, point P. The identity S= Rθ and dS = Rdθ may be of use.




14. A total charge Q is distributed uniformly along a straight rod of length L. Find the potential at a point P at a distance h from the midpoint of the rod. (Hint: ò[x²+k²]^-½dx = ln[x+[x²+k²]^½] + C). Use the gradient with respect to h to find the electric field at that point.




15. Three thin rods of glass of length L carry charges uniformly distributed along their lengths. The charges on the three rods are +Q, +Q, and -Q, respectively. The rods are arranged along the sides of an equilateral triangle. What is the electrostatic potential at the midpoint of this triangle? (Hint: Use the result of the question 20-7.)




16. The electric potential over a certain region is given by V = 3x²y-4xz-5y² volts. Determine the components of the electric field and evaluate at the point (+1,0,+2).
17. Over a certain region of space, the electric potential is V = 5x-3x²y+2yz². Determine the components of the electric field and evaluate electric field at the point (1,0,-2).

Questions?mike.coombes@kwantlen.ca