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Three point charges, which initially are infinitely far apart, are placed at the corners of an equilateral triangle with sides d. Two of the point charges are identical and have charge q. If zero net work is required to place the three charges at the corners of the triangle, what must the value of the third charge be?

Short Answer

Expert verified

The value of the third charge must be -q2.

Step by step solution

01

Potential energy

Potential energy is given by:

U=ki<jqiqjrij

Where k is a constant and the system is made of n charges as q1,q2,q3,q4................qn,

and the distance between these charges are r12(distancebetweenr1andr2),r23(distancebetweenr2andr3),r34(distancebetweenr3andr4),r45(distancebetweenr4andr5).........

02

Determine the third charge

The potential energy of the system is:

U=ki<jqiqjrij

From the given statement, the diagram can be drawn as:

For three charge system, the above expression can be written as:

U=kq1q1r12+q2q3r23+q1q3r13

The charges q1=q2=qandq3=qcand distances r12=r23=r13=r.

Thus,

U=kqqr+qqcr+qqcr=k=qqr+2qqcr

The work done to create the system is stored as its potential energy. For work done to be zero:

W=00=Ukqqr+2qqcr=0

Solving further as:

qqr+2qqcr=0q2+2qqc=0q2=-2qqcqc=-q2

Therefore, the third charge is-q2

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