91影视

Consider the formula for the electric field

$E=\frac{1}{4蟺{蔚}_0}\frac{Q}{r^2}$

Here${蔚}_0$, is the permittivity of the free space,$Q$ is the charge and$r$ is the distance between the sphere.

Consider the expression for the current is

$\mathbf{I}\mathbf{=}\frac{V}{R}$

Determine the electric filed between concentric metal spheres.

$E=\frac{1}{4蟺{蔚}_0}\frac{Q}{r^2}$

If the voltage potential difference is Vin the concentric spheres having radius aand b.

Write the expression for the voltage difference as

$\begin{array}{c}{V}_a鈭�V_b=鈭�{鈭�}_b^a\frac{Q}{4蟺{蔚}_0}\frac{1}{r^2}dr\\ =鈭�\frac{Q}{4蟺{蔚}_0}{鈭�}_b^a\frac{1}{r^2}dr\\ =\frac{Q}{4蟺{蔚}_0}(\frac{1}{a}鈭�\frac{1}{b})\end{array}$ 鈥�.. (1)

Consider the formula for the electric current in terms of the electric current density is

$\begin{array}{c}I=蟽鈭�E鈰�da\\ =蟽\frac{Q}{{蔚}_0}\end{array}$

From equation (1) rewrite the expression for current as

.$\begin{array}{l}I=\frac{蟽4蟺{蔚}_0(V_a鈭�V_b)}{{蔚}_0(\frac{1}{a}鈭�\frac{1}{b})}\\ I=\frac{蟽4蟺(V_a鈭�V_b)}{(\frac{1}{a}鈭�\frac{1}{b})}\end{array}$

Therefore, the expression for the current is$I=\frac{蟽4蟺(V_a鈭�V_b)}{(\frac{1}{a}鈭�\frac{1}{b})}$ .

Consider the formula for the resistance as

$R=\frac{V}{I}$

Rewrite the expression for the resistance in terms of the voltage difference as

$\begin{array}{c}R=\frac{V_a鈭�V_b}{\frac{蟽4蟺(V_a鈭�V_b)}{(\frac{1}{a}鈭�\frac{1}{b})}}\\ =\frac{1}{4蟺蟽}(\frac{1}{a}鈭�\frac{1}{b})\end{array}$

Consider that b>>>a here, on negating athe sphere feel current by the sphere b on the basis of the difference between both the sphere. The expression is鈥�

$R=\frac{1}{4蟺蟽a}$

Since, the resistance is due to the inner sphere, the successive shells have less contribution in the current because of the small cross sectional area.

Write the expression for the two submerged sphere as

$\begin{array}{c}R=\frac{2}{4蟺蟽a}\\ =\frac{1}{2鈮�蟺蟽a}\end{array}$

From the general expression for the resistance solve as

$\begin{array}{c}R=\frac{V}{I}\\ I=\frac{V}{\frac{1}{2蟺蟽a}}\\ I=2V蟺蟽a\end{array}$

Therefore, the expression for the current between the two sphere is $2V蟺蟽a$.