Dr. Huerta
E and B fields of moving charges
Phy 210
Charge with velocity v and acceleration a
A charge q moving with velocity v and acceleration a as shown in the figure to the left below
produces an electric field E(r, t) and a magnetic field B(r, t) at a position r, at time t due to the
position velocity and acceleration of the charge at an earlier time t0 = t − r/c (retarded time)
because the information travels at the speed of light c.
E(r,t)
v
q
r
B(r,t)
a
position at t'=t-r/c
q
1
E(r, t) =
4π0 |r − r · v/c|3
rv v2 r
r−
1− 2 + 2 ×
c
c
c
rv r−
×a
c
and B =
r×E
rc
(1)
The E field of a fast moving charge shown in the figure to the left below is concentrated in a
direction perpendicular to the velocity v. Remarkably E points away from the present position of
the charge, which is extraordinary because the “message” came from the retarded position. Also
notice that if a 6= 0, v c, and r is very large the magnitude of E varies as 1/r, which is the
radiation field Erad produced by an accelerating charge, and
E(r, t)rad =
q 1 r × (r × a)
4π0 r3
c2
(2)
Electric dipole antenna The figure to the right above shows an oscillating electric dipole antenna.
of length d. To produce a wave of wavelength λ = fc = 2πω c a good length for the antenna is
d = λ/2, so each arm is λ/4.
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Say the oscillating electric dipole at the origin of the spherical coordinate system in the figure
above, where at the point shown φ̂ is into the paper, is p(t) = p0 cos ωt ẑ where p0 = q0 d and d is
the length of the antenna. Assuming that d λ and d r, the E and B fields generated by that
antenna, where t0 = t − r/c is the retarded time, are
p0 sin θ
2 p0 cos θ cos ωt0 ω sin ωt0
1
ω2 ω sin ωt0
0
, Eθ (r, θ, φ, t) =
Er (r, θ, φ, t) =
−
−
cos ωt −
4π0
r3
r2 c
4π0
r3 rc2
r2
2
−µ0 p0 sin θ ω cos ωt0 ω sin ωt0
, with Eφ = 0, Br = 0, and Bθ = 0.
Bφ (r, θ, φ, t) =
−
4π
rc
r2 c
(3)
Since cos ωt0 = cos(ωt − ωr/c) and sin ωt0 = sin(ωt − ωr/c), the fields are waves traveling in the
radial direction with speed c. In the radiation zone, at large distances from the antenna, where
r λ, terms in 1/r2 and in 1/r3 are negligible compared to the radiation terms that depend on
1/r. So for r λ the radiation fields have Erad in the θ̂ direction and Brad in the φ̂ direction as
follows
Erad ≈
−p0 ω 2 sin θ cos ωt0
−µ0 p0 ω 2 sin θ cos ωt0
r × Erad
θ̂
and
B
(r,
θ,
φ,
t)
=
φ̂ =
.
rad
2
4π0 c r
4πc r
c
(4)
We see that the fields are strongest at θ = 90◦ and zero at θ = 0◦ . The Poynting vector S =
far away from the antenna in the radiation region is in the radial direction with
2
µ0 p0 ω 2 sin θ
µ0 p20 ω 4 sin2 θ
S=
cos ω(t − r/c) r̂ so Saverage =
r̂,
c
4πr
32π 2 c r2
E×B
µ0
(5)
so no energy goes in the direction θ = 0◦ , and the profile of intensity looks like a donut as shown in
the figure at the center below. A vertical dipole antenna is used to detect the electric field radiated
by another vertical dipole antenna, however it is not directional and the receiver cannot find out
where the waves are coming from. To detect where the transmitted waves are coming from we need
a magnetic dipole loop antenna, which is directional.
z
φ
r
θ
θ
i(t)
a µ
r
y
φ
x
Magnetic dipole (loop) antenna
The figure to the right above shows an oscillating magnetic dipole antenna. It is a loop of radius a
with a current i(t) = I0 cos ωt so the antenna has a magnetic dipole moment µ̂(t) = m0 cos ωt ẑ in
the z direction where m0 = πa2 I0 . We consider a small antenna, for which a λ/2π, or aω/c 1.
The E and B fields in the radiation zone, where r λ are
Erad ≈
µ0 m0 ω 2 sin θ cos ω(t − r/c)
µ0 m0 ω 2 sin θ cos ω(t − r/c)
r × Erad
φ̂ and Brad ≈
θ̂ =
. (6)
2
4πc r
4πc r
c
2
The fields are ”duals” of the ones in the electric dipole antenna. The Poynting vector S =
the radiation region is again in the radial direction with
2
µ0 m0 ω 2 sin θ
µ0 m20 ω 4 sin2 θ
S=
cos ω(t − r/c) r̂ so Saverage =
r̂.
c
4πr
32π 2 c r2
E×B
µ0
in
(7)
The ratio of Poynting vectors
dipole
|Smagnetic
|
average
dipole
|Selectric
|
average
=
m0
p0 c
2
.
(8)
For purposes of reasonable comparison
let p0 =
dipole
I0 a
|Smagnetic
|
average
=
, then
electric dipole
ω
|Saverage
|
aω
c
2
1.
(9)
So the electric dipole antenna is a much better emitter of EM waves than the magnetic dipole
antenna. However the magnetic dipole antenna is a useful directional receiver of electromagnetic
waves. Say a vertical electric dipole antenna is sending waves. A vertical loop detects the flux due
to a horizontal magnetic field. Rotating the loop around a vertical axis can determine where the
wave is coming from and locate the transmitter.
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