(a) Two point charges + Q1 and – Q2 are placed r distance apart. Obtain the expression for the amount of work done to place a third charge Q3 at the midpoint of the line joining the two charges (b) at what distance from charge + Q1 , on the line joining the two charges (in terms of Q1, Q2 and r) will this work done be zero.
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12th
Physics
Electrostatic Potential and Capacitance
Potential Energy of a System of Charges
Two point charges 2q and ...
PHYSICS
Two point charges 2q and 8q are placed at a distance r apart. Where should a third charge −q be placed between them, so that the electrical potential energy of the system is minimum?
October 15, 2019
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Tajkera Momin
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ANSWER
Let the third charge −q is placed at a distance of x from the charge 2q. Then, the potential energy of the system is
U=k[
r
2q(8q)
−
x
(2q)(q)
−
r−x
(8q)(q)
]
Here, k=
4πε
0
1
U is minimum, where
x
2
+
r−x
8
is maximum
Let
x
2
+
r−x
8
=y
For y to be maximum,
dx
dy
=0
−
x
2
2
+
(r−x)
2
8
=0
⇒
r−x
x
=
8
2
=
2
1
⇒x=
3
r
But at x=
3
r
,
dx
2
d
2
y
= positive i.e. at x=
3
r
, y is minimum or U is maximum. So, there cannot be any point between them where the potential energy is minimum.