Answer to Question #118564 in Electricity and Magnetism for Pieter Coetze

Question #118564
Two spheres, P and Q with charges of +4 microcoulomb and -6 microcoulomb are brought into contact and separated again.
What is the charge on each sphere after contact?
1
Expert's answer
2020-05-28T13:07:27-0400

The potential of the first sphere before after contact is:


φ1=14πε0q1R1\varphi_1' = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_1'}{R_1}

where q1q_1' is the charge on the first sphere after contact and R1R_1 is the radius of the first sphere.

The potential of the second sphere after contact is:


φ2=14πε0q2R2\varphi_2' = \dfrac{1}{4\pi\varepsilon_0}\dfrac{q_2'}{R_2}

where q2q_2' is the charge on the second sphere after contact and R2R_2 is the radius of the second sphere.

As far as the charge conserves, the total anoum of charge will not change:

q1+q2=4μC+(6μC)=2μCq_1' + q_2' = 4\mu C + (-6\mu C) = -2\mu C


After contact the potentials of the spheres will be equal to each other:


φ1=φ2q1R1=q2R2\varphi_1' = \varphi_2' \Rightarrow \dfrac{q_1'}{R_1} = \dfrac{q_2'}{R_2}

Express q1q_1' and substitute it to the previous equation:

q1=q2R1R2q2R1R2+q2=2μCq2=2R2R1+R2μCq_1' = q_2'\dfrac{R_1}{R_2}\\ q_2'\dfrac{R_1}{R_2} + q_2' = -2\mu C\\ q_2' = -2 \dfrac{R_2}{R_1 + R_2} \mu C

The charge of the first sphere will be:


q1=2R1R1+R2μCq_1' = -2 \dfrac{R_1}{R_1 + R_2} \mu C


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