A small sphere of radius r₁ and charge q₁ is enclosed by a spherical shell of radius r₂and charge q₂. Show that if q₁ is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge q₂ on the shell is.

Solution

Potential of inner sphere due to charge q_1-

V_{1} = \frac{1}{4 {πε}_{0}} \frac{q_{1}}{r_{1}}

Potential of inner sphere due to the enclosed sphere-

V_{2} = \frac{1}{4 {πε}_{0}} \frac{q_{2}}{r_{2}}

Thus, total potential of inner sphere = V₁ + V₂

V = \frac{1}{4 {πε}_{0}} (\frac{q_{1}}{r_{1}} + \frac{q_{2}}{r_{2}})

Potential of shell-

V' = \frac{1}{4 {πε}_{0}} \frac{q_{2}}{r_{2}}

Potential difference between inner sphere and shell= V – V’

= \frac{1}{4 {πε}_{0}} (\frac{q_{1}}{r_{1}} + \frac{q_{2}}{r_{2}}) - \frac{1}{4 {πε}_{0}} \frac{q_{2}}{r_{2}} = \frac{1}{4 {πε}_{0}} \frac{q_{1}}{r_{1}}

Therefore, q₁ is positive as seen from the above equation.

Since, from the above equation, potential difference does not depend on q₂ we can conclude that, the charge will always flow from the sphere to shell, no matter whatsoever charge is on the shell.

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A small sphere of radius r1 and charge q1 is enclosed by a spherical shell of radius r2 and charge q2. Show that if q1 is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge q2 on the shell is.

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A small sphere of radius r₁ and charge q₁ is enclosed by a spherical shell of radius r₂and charge q₂. Show that if q₁ is positive, charge will necessarily flow from the sphere to the shell (when the two are connected by a wire) no matter what the charge q₂ on the shell is.