Introductory Physics Volume Two

48 Electric Potential 2.6
⊲ Problem 2.13
A 120µF capacitor is charged to an electric potential difference of 100V.
(a) How much energy is stored in the capacitor?
(b) If the field strength in a capacitor becomes to great then the charge
will jump across the gap between the plates. Assume that this break
down occurs when the field strength reaches 3 × 10 6 V m. What is the
minimum volume that is required for a 120µF capacitor to be able to
hold an electric potential difference of 100V?
§ 2.6 Electric Potential of a Point Charge
We wish to find the electric potential for a point charge. The
electric field strength around a point charge is
E =
q 1
4πɛ 0 r 2
where r is the distance from the point charge. The electric field is
either pointed away or toward the charge depending on if the charge is
positive or negative. Let us start with the positive charge.
V b − V a = −
∫ rb
r a
⃗ E · ⃗ dr
If we let r b > r a then dr ⃗ is pointed outward. This is the same direction
as E ⃗ so that E ⃗ · ⃗dr = E dr. Thus
∫ rb
V b − V a = − E dr = − q ∫ rb
1
r a
4πɛ 0 r a
r 2 dr = − q [− 1 ] rb
4πɛ 0 r
r a
= q 1
−
q 1
4πɛ 0 r b 4πɛ 0 r a
Thus we see that the electric potential at a distance r from a point
charge is
V (r) =
q 1
4πɛ 0 r + constant
For simplicity one normally chooses for the constant to be zero. Note
that this choice for the constant implies that the zero of the electric
potential is at the position r = ∞.
Theorem: Electric Potential of a Point Charge
V (r) =
q 1
4πɛ 0 r
⊲ Problem 2.14
Show that V (r) =
q 1
4πɛ 0 r
is correct for a negative charge as well.