An electricity and magnetism problem by A Former Brilliant Member

Electricity and Magnetism Level 2

Two metal spheres of radii r 1 , r 2 r_1 , r_2 such that r 1 < r 2 r_1<r_2 are very far apart but connected by a thin wire. If their combinated charge is Q Q the what is their common potential?

k = 1 4 π ϵ 0 N m 2 C − 2 k = \dfrac {1}{4π\epsilon_0}Nm^2C^{-2}

− k Q ( r 1 + r 2 ) -\dfrac {kQ}{(r_1 + r_2)} − k Q r 1 r 2 -\dfrac {kQ}{r_1 r_2} k Q ( r 1 + r 2 ) \dfrac {kQ}{(r_1 + r_2)} k Q ( r 1 − r 2 ) \dfrac {kQ}{(r_1 - r_2)}

A Former Brilliant Member by A Former Brilliant Member

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1 solution

As they are far apart, k q 1 r 1 = k q 2 r 2 \dfrac {kq_1}{r_1} = \dfrac {kq_2}{r_2}

⇒ q 1 r 1 = q 2 r 2 \Rightarrow \dfrac {q_1}{r_1 } = \dfrac {q_2}{r_2 }

Also, q 1 + q 2 = Q q_1 + q_2 = Q

We get, q 1 = Q r 1 ( r 1 + r 2 ) q_1 = \dfrac {Qr_1}{(r_1 + r_2)}

Hence, potential difference V = k Q ( r 1 + r 2 ) V =\boxed{ \dfrac {kQ}{(r_1 + r_2)}}

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