The cross-sectional area and length of cylindrical conductor are A and l respectively. The specific conductivity varies as σ(x)=σ0l/x where x is the distance along the axis of the cylinder from one of ends. A potential difference V0 is applied across the cylinder. Choose the correct statements

i) dR=1σ(x)dxA=xdxAlσ0 ∴R=∫0ldR=1Alσ0∫0lxdx=2l3Aσ0ii) v0R=3Aσ0v02lThe current density, J=IA=3σ0v02lElectric field, E(x)=Jσ(x)=3V0x2l3/2

The cross-sectional area and length of cylindrical conductor are A and l respectively. The specific conductivity varies as σ(x)=σ0l/x where x is the distance along the axis of the cylinder from one of ends. A potential difference V0 is applied across the cylinder. Choose the correct statements

i) dR=1σ(x)dxA=xdxAlσ0 ∴R=∫0ldR=1Alσ0∫0lxdx=2l3Aσ0ii) v0R=3Aσ0v02lThe current density, J=IA=3σ0v02lElectric field, E(x)=Jσ(x)=3V0x2l3/2

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The cross-sectional area and length of cylindrical conductor are A and l respectively. The specific conductivity varies as $σ (x) = σ_{0} l / \sqrt{x}$ where x is the distance along the axis of the cylinder from one of ends. A potential difference $V_{0}$ is applied across the cylinder. Choose the correct statements

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The resistance of the system along the cylindrical axis is $\frac{2 \sqrt{l}}{3 A σ_{0}}$

The current density is constant through out the length of the cylinder.

The electric field E(x) at each point in the cylinder is $E (x) = \frac{5 V_{0} \sqrt{x}}{2 l^{3 / 2}}$

The electric field E(x) at each point in the cylinder is $E (x) = \frac{3 V_{0} \sqrt{x}}{2 l^{3 / 2}}$

answer is A, B, D.

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Detailed Solution

i) $d R = \frac{1}{σ (x)} \frac{d x}{A} = \frac{\sqrt{x} d x}{A l σ_{0}}$
$∴ R = \int_{0}^{l} d R = \frac{1}{A l σ_{0}} \int_{0}^{l} \sqrt{x} d x = \frac{2 \sqrt{l}}{3 A σ_{0}}$
ii) $\frac{v_{0}}{R} = \frac{3 A σ_{0} v_{0}}{2 \sqrt{l}}$
The current density, $J = \frac{I}{A} = \frac{3 σ_{0} v_{0}}{2 \sqrt{l}}$
Electric field, $E (x) = \frac{J}{σ (x)} = \frac{3 V_{0} \sqrt{x}}{2 l^{3 / 2}}$