SME
Electromagnetic Theory · Problem type
The potential of an electric dipole, far away
Use this for “Charges of +3 nC and −3 nC sit 2 mm apart. Find the potential 5 cm from their center, 45° from the axis on the + side.”
Diagram
p
=
qs
=
1 ×
10
-9
C
×
0.001
m
p
=
1 ×
10
-12
C·m
V
=
kp
cos θ
r
2
V
=
8.99 ×
10
9
×
1 ×
10
-12
×
cos
60°
÷
0.1
2
V = 0.4495 V
Exactly, kq(1/r₊ − 1/r₋) = 0.4495 V: the far-field form is close because r ≫ s.
V is positive on the + side, 0 across the middle (θ = 90°) and negative on the − side; s is drawn larger than to scale.
This is an example. Type your own values to start a new problem.
k
q (
)
s (
)
cos
°
(
r (
)
)
2
=
V (
)
Formulas
V
=
8.99 ×
10
9
×
q
×
s
×
cos(
θ
)
r
2
+0.4495
=
8.99 ×
10
9
×
(1
×
10
-9
)
×
0.001
×
cos(60°)
0.1
2
Formulas are worked in C, m. The step-by-step shows the conversions.
Assumptions
•
Charges +q and −q sit s apart; the dipole moment p = qs points from −q to +q. k = 8.99 × 10⁹ N·m²/C².
•
P is r from the center at θ from the dipole’s axis; r is far larger than s (here at least 10 times), so the two distances differ by s cos θ.
•
Then V = kq ÷ r₊ − kq ÷ r₋ ≈ kqs cos θ ÷ r² = kp cos θ ÷ r²: positive on the + side, 0 across the middle (θ = 90°), negative on the − side.
Step-by-step
WE KNOW
q = 1 nC, s = 1 mm, r = 10 cm, θ = 60°
FIND
potential at P (V)
Step 1 · Convert to the formula’s units
The formulas don’t work directly in the units you chose, so convert first (to C, m).
q = 1 nC = 1 × 10⁻⁹ C (1 C = 10⁹ nC)
s = 1 mm = 0.001 m (1 m = 1000 mm)
r = 10 cm = 0.1 m (1 m = 100 cm)
Step 2 · Find potential at P (V)
Use V = 8.99 × 10⁹ × q × s × cos(θ) ÷ r²
kp
cos
θ
r
2
with
p
=
qs:
the
two
charges’
potentials
nearly
cancel,
and
what
is
left
falls
as
1
r
2
.
V
=
8.99 ×
10
9
×
10
-9
×
0.001
×
cos(60°)
0.1
2
V
=
(8.99
×
10
9
)
×
10
-9
×
0.001
×
0.5
0.01
V
=
8.99
×
0.001
×
0.5
0.01
V
=
0.00899
×
0.5
0.01
V
=
0.004495
0.01
V
=
+0.4495
V
Check
Put the numbers back in. Both sides should match.
+0.4495
=
8.99 ×
10
9
×
10
-9
×
0.001
×
cos(60°)
0.1
2
✓
Topic check
See what you know: 5 questions on this topic
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