5 of 6 Constants A capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has radius 10.0 cm, and the outer sphere has radius 14.5 cm. A potential difference of 110 V is applied to the capacitor.

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Question:

A spherical capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has radius 10.0 cm and the outer sphere has radius 14.5 cm. A potential difference of 110 V is applied to the capacitor.

(a) What is the energy density at r = 10.1 cm, just outside the inner sphere? in J/m3

(b) What is the energy density at r = 14.4 cm, just inside the outer sphere? in J/m3

Answer:

U = 0.0218 j/m³

U = 0.00527 j/m³

Given Information:

Radius inner sphere = ri = 10 cm = 0.10 m

Radius outer sphere = ro = 14.5 cm = 0.145 m

Potential difference = V = 110 V

E = 8.85x10⁻¹² C²/N. m²

Required Information:

Energy density U = at r = 10.1 cm = 0.101 m

Energy density U = at r = 14.4 cm = 0.144 m

Explanation:

(a) What is the energy density at r = 10.1 cm, just outside the inner sphere?

The capacitance can be found by

Capacitance = 4πEo (ri*ro/ri+ro)

Capacitance = 4π8.85x10⁻¹² (0.10*0.145/0.10+0.145)

Capacitance = 6.581x10⁻¹² F

The charge on the capacitor can found using

Charge = Q = CV

Q = 6.581x10⁻¹²*110 = 7.964x10⁻⁸

The Electiric field is given by

E = 1/4πEo (Q/r²)

where r = 0.101 m

E = 1/4π8.85x10⁻¹² (7.964x10⁻⁸ / (0.101²)

E = 70199.7 V/m

Energy density is given by

U = 1/2EoE²

U = 1/2*8.85x10⁻¹² (70199.7) ²

U = 0.0218 j/m³

(b) What is the energy density at r = 14.4 cm, just inside the outer sphere?

The Electiric field is given by

E = 1/4πEo (Q/r²)

where r = 0.144 m

E = 1/4π8.85x10⁻¹² (7.964x10⁻⁸ / (0.144²)

E = 34534.5 V/m

Energy density is given by

U = 1/2EoE²

U = 1/2*8.85x10⁻¹² (34534.5) ²

U = 0.00527 j/m³