Question

What is the potential difference per unit length between two infinitely long concentric cylindrical shells with inner radius 1.5 cm and outer radius 5.6 cm, if the charge on the inner and outer cylinders are 7.0 nC and -7.0 nC, respectively

Answer 1

Answer:

165.8 V/m

Explanation:

The capacitance of a long concentric cylindrical shell of length, L and inner radius, a and outer radius, b is C = 2πε₀L/㏑(b/a)

Since the charge on the cylindrical shells, Q = CV where V = the potential difference across the capacitor(which is the potential difference between the concentric cylindrical shells)

V = Q/C

V = Q ÷ 2πε₀L/㏑(b/a)

V = Q㏑(b/a)/2πε₀L

So, the potential difference per unit length V' is

V' = V/L = Q㏑(b/a)/2πε₀

Given that a = inner radius = 1.5 cm, b = outer radius = 5.6 cm and Q = 7.0 nC = 7.0 × 10⁻⁹ C and ε₀ = 8.854 × 10⁻¹² F/m substituting the values of the variables into the equation, we have

V' = Q㏑(b/a)/2πε₀

V' = 7.0 × 10⁻⁹ C㏑(5.6 cm/1.5 cm)/(2π × 8.854 × 10⁻¹² F/m)

V' = 7.0 × 10⁻⁹ C㏑(3.733)/(55.631 × 10⁻¹² F/m)

V' = 7.0 × 10⁻⁹ C × 1.3173/(55.631 × 10⁻¹² F/m)

V' = 9.2211 × 10⁻⁹ C/(55.631 × 10⁻¹² F/m)

V' = 0.16575 × 10³ V/m

V' = 165.75 V/m

V' ≅ 165.8 V/m