Two charges Q1=q and Q2=mq are placed at the points P1(a,b) and P2(ma,mb), respectively, in the XY plane, where a,b 0 and m 0,1. If V1 is the potential at a point in the XY plane due to charge Q1 and V2 is the potential at that point due to charge Q2. Correct statement(s) for the points at which |V1|=|V2| is/are:
- For m=-1, locus of these points is ax+by=0.
- For m=2, the locus of these points is a circle of radius 23 a2+b2 centered at ( 23a, 23b )
- For m=-2, the locus of these points is a circle of radius 2 a2+b2 centered at (2a,2b)
- For m=-3, locus of these points is 3bx+3ay=0.
Answer
(1) For m=-1, locus of these points is ax+by=0., (2) For m=2, the locus of these points is a circle of radius 23 a2+b2 centered at ( 23a, 23b ), (3) For m=-2, the locus of these points is a circle of radius 2 a2+b2 centered at (2a,2b)
Solution
Key idea: |V1|=|V2| |q|r1= |m||q|r2 r22=m2r12:
(x-ma)2+(y-mb)2=m2 [(x-a)2+(y-b)2 ].
m=-1: (x+a)2+(y+b)2=(x-a)2+(y-b)2 ax+by=0. (A) true.
m=2: 3x2+3y2-4ax-4by=0, centre ( 2a3, 2b3 ), radius 23 a2+b2. (B) true.
m=-2: 3x2+3y2-12ax-12by=0 x2+y2-4ax-4by=0, centre (2a,2b), radius 2 a2+b2. (C) true.
m=-3: 8x2+8y2-24ax-24by=0 is a circle, not a line. (D) false.
Answer: (A), (B), (C)