Answer:
Explanation:
To determine the new energy density when the charge is doubled, we first need to understand the relationship between the energy density in the electric field and the charge.
The energy density
u in an electric field due to a point charge
Q at a distance
d is given by:
=
0
2
2
u=
2
ϵ
0
E
2
where
E is the electric field strength and
0
ϵ
0
is the permittivity of free space.
The electric field
E due to a point charge
Q at a distance
d is:
=
4
0
2
E=
4πϵ
0
d
2
Q
When the charge
Q is doubled to
2
2Q, the new electric field
′
E
′
at the same distance
d is:
′
=
2
4
0
2
=
2
(
4
0
2
)
=
2
E
′
=
4πϵ
0
d
2
2Q
=2(
4πϵ
0
d
2
Q
)=2E
The new energy density
′
u
′
will be:
′
=
0
(
′
)
2
2
=
0
(
2
)
2
2
=
0
⋅
4
2
2
=
4
(
0
2
2
)
=
4
u
′
=
2
ϵ
0
(E
′
)
2
=
2
ϵ
0
(2E)
2
=
2
ϵ
0
⋅4E
2
=4(
2
ϵ
0
E
2
)=4u
Therefore, if we double the charge
Q, the energy density at the same point will be:
4
4u