Charges In Equilibrium

Electricity and Magnetism Level 2

Three charged particles are placed in such a way that one of the particles is the midpoint of a line segment connecting the other two. If the particles at the ends of the line segment have charge Q Q , then what should the the charge of the third particle be so that the system is in equilibrium?

Q 4 \frac{Q}{4} Q 2 \frac{Q}{2} − Q 4 -\frac{Q}{4} − Q 2 -\frac{Q}{2}

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Samara Simha Reddy by Samara Simha Reddy , 22, India

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1 solution

Tom Engelsman
Feb 21, 2021

Let us place the two + Q +Q charges on the x − x- axis at ( − r , 0 ) (-r,0) and ( r , 0 ) (r,0) . If the third charge is positive, then it will repel when placed at the midpoint ( 0 , 0 ) (0,0) of the aforementioned positive Q-charges. It must be negatively charged to allow equilibrium in this system (call it − q ) -q) . By Coulomb's Law ( F = k q 1 q 2 d 2 F = \frac{kq_{1}q_{2}}{d^2} ), the attractive force between Q , − q Q, -q must equal the repulsive force between Q , Q Q,Q . We require:

k Q 2 ( 2 r ) 2 = − k Q q r 2 ⇒ q = − Q 4 . \frac{kQ^2}{(2r)^2} = -\frac{kQq}{r^2} \Rightarrow \boxed{q = -\frac{Q}{4}}.

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