Consider a linear antenna of length d (d << λ)
with a narrow gap in the center for the purposes of excitation. Assume
that the current is sinusoidal and in the same direction in each half of the
antenna, having a value of I_{0} at the gap and falling linearly to zero
at the ends. Find the power radiated in the electric dipole approximation.

Solution:

- Concepts:

Radiation from antennas, dipole radiation - Reasoning:

We can treat the antenna as made up of small sections, each section emitting dipole radiation. - Details of the calculation:

Consider a small section of the antenna of length dz' located at z' . Treat it like a dipole.

Let p(z') = qdz'(**z**/z), q(z') = q_{0}(z')cosωt.

Then I(z') = dq(z')/dt = -ωq_{0}(z')sinωt = I_{0}(z')sin(ωt), I_{0}(z') = -ωq_{0}(z').

p(z') = p_{0}(z')cosωt. p_{0}(z') = q_{0}(z')dz = -(I_{0}(z')/ω)dz.

For dipole at the origin we have**E**_{R}(**r**,t) = -(1/(4πε_{0}c^{2}r))ω^{2}p_{0}cos(ω(t-r/c))sinθ (**θ**/θ).

A small section of the antenna of length dz' located at z' a distance**R**=**r**- (**z**/z)z' from the observation point therefore produces

d**E**_{R}(**r**,t) = (1/(4πε_{0}c^{2}R))ω^{2}(I_{0}(z')/ω)cos(ω(t-R/c))sinθ'dz' (**θ'**/θ').

d**B**_{R}(**r**,t) = (1/(4πε_{0}c^{3}R))ω^{2}(I_{0}(z')/ω)cos(ω(t-R/c))sinθ'dz'(**φ**/φ).

Assume r >> λ, r >> z', d << λ.

Then θ' ≈ θ = constant, for all sections of the antenna.

We can also replace 1/R by 1/r and cos(ωt - kR) by cos(ωt - kr) because the changes in kR are small compared to π/2 over the length of the antenna. (k = ω/c)

I_{0}(z) = I_{0}- (2I_{0}/d)|z|, the current falls linearly to zero at the ends.

E_{θ}(**r**,t) = ∫_{z=-d/2}^{z=d/2}dE_{θ}(**r**,t) = 2 ∫_{0}^{z=d/2}dE_{θ}(**r**,t)

= (I_{0}ω/(2πε_{0}c^{2}r))cos(ω(t-r/c))sinθ ∫_{0}^{z=d/2}(1-2z/d)dz'

= (I_{0}dω^{2}/(8πε_{0}c^{3}))(1/(kr))cos(ω(t-r/c))sinθ.

B_{φ}(**r**,t) = ∫_{z=-d/2}^{z=d/2}dB_{φ}(**r**,t) = 2 ∫_{0}^{z=d/2}dB_{φ}(**r**,t)

= (μ_{0}I_{0}dω^{2}/(8πc^{2}))(1/(kr))cos(ω(t-r/c))sinθ.

**S**= (1/μ_{0})**E**×**B**= (**r**/r)[ (I_{0}^{2}d^{2}ω^{4}/(64π^{2}ε_{0}c^{5}k^{2}r^{2}))cos^{2}(ω(t-r/c))sin^{2}θ.]

P = (I_{0}^{2}d^{2}ω^{2}/(64π^{2}ε_{0}c^{3}r^{2}))cos^{2}(ω(t-r/c)) ∫2πr^{2}sin^{3}θ dθ

= (I_{0}^{2}d^{2}ω^{2}/(32πε_{0}c^{3}))cos^{2}(ω(t-r/c))(4/3).

<P> = ½(I_{0}^{2}d^{2}ω^{2}/(24πε_{0}c^{3})) is the average power radiated in the electric dipole approximation.

Assume an antenna is aligned with the z-axis and extends
from z = -l/4 to z = +l/4. Assume the current in the antenna varies as I(z,t) = I_{0}sin(ωt)cos(kz).
Calculate the radiation fields E_{R} and B_{R} and the average
Poynting vector.

Solution:

- Concepts:

Radiation from antennas, dipole radiation - Reasoning:

We can treat the antenna as made up of small sections, each section emitting dipole radiation. - Details of the calculation:

Consider a small section of the antenna. Treat it like a dipole.

Let p = qdz(**z**/z), q = q_{0}cosωt.

Then I = dq/dt = -ωq_{0}sinωt = I_{0}sin(ωt), I_{0}= -ωq_{0}.

p = p_{0}cosωt. p_{0}= q_{0}dz = -(I_{0}/ω)dz.

For dipole at the origin we have**E**_{R}(**r**,t) = -(1/(4πε_{0}c^{2}r))ω^{2}p_{0}cos(ω(t-r/c))sinθ (**θ**/θ).

A small section of the antenna of length dz' located at z' a distance**R**=**r**- (**z**/z)z' from the observation point therefore produces

d**E**_{R}(**r**,t) = (1/(4πε_{0}c^{2}R))ω^{2}(I_{0}/ω)cos(ω(t-R/c))sinθ'cos(kz')dz' (**θ**/θ).

d**B**_{R}(**r**,t) = (1/(4πε_{0}c^{3}R))ω^{2}(I_{0}/ω)cos(ω(t-R/c))sinθ'cos(kz')dz'(**φ**/φ).

dE_{θ}(**r**,t) = (ω^{2}I_{0}/(4πε_{0}c^{3}))[cos(ω(t-R/c))/kR]sinθ'cos(kz')dz',

dB_{f}(**r**,t) = (ω^{2}I_{0}/(4πε_{0}c^{4}))[cos(ω(t-R/c)/kR]sinθ'cos(kz')dz',

with k = ω/c.

E_{θ}(**r**,t) = (ω^{2}I_{0}/(4πε_{0}c^{3}))∫[cos(ω(t-R/c))/kR]sinθ'cos(kz')dz',

B_{f}(**r**,t) = (ω^{2}I_{0}/(4πε_{0}c^{4}))∫[cos(ω(t-R/c)/kR]sinθ'cos(kz')dz'.

The integration limits are -l/4 and l/4.

Assume r >> l, r >> z'.

Then θ' ≈ θ = constant for all sections of the antenna.

But cos(ωt - kR) ≠ cos(ωt - kr) because kR changes by π/2 over the length of the antenna. Although kR >> π/2, this changes the phase drastically.

We use kR = kr - kz'cosθ for the argument of the cosine function. We, however, can safely replace 1/R by 1/r.

We then have

E_{θ}(**r**,t) = (ω^{2}I_{0}sinθ/(4πε_{0}c^{3}kr))∫cos(ωt-kr+kz'cosθ)cos(kz')dz',

B_{f}(**r**,t) = (μ_{0}ω^{2}I_{0}sinθ/(4πc^{2}kr))∫cos(ωt-kr+kz'cosθ)cos(kz')dz'.

The integration limits are -l/4 and l/4.

∫_{-l/4}^{l/4}cos(ωt-kr+kz'cosθ)cos(kz')dz'

= (1/k)cos(ωt-kr)∫_{-π/2}^{π/2}cos(kz'cosθ)cos(kz')dkz'

- (1/k)sin(ωt-kr)∫_{-π/2}^{π/2}sin(kz'cosθ)cos(kz')dkz'.

The second integral is zero, since it is an integral over an odd function of z'.

∫_{-π/2}^{π/2}cos(ax)cos(x)dx = sin((a-1)x)/(2(a-1)) + sin((a+1))x/(2(a+1))|_{-π/2}^{π/2}

= [1/(2(a-1))]{sinaxcosx-sinxcosax} + [1/(2(a+1))]{sinaxcosx+sinxcosax}|_{-π/2}^{π/2}

= [1/(2(a-1))]{-2cos(a(π/2))} + [1/(2(a+1))]{2cos(a(π/2))}

= [1/(2(a^{2}-1))]{-4cos(a(π/2))}

Wit a = cosθ ωe have

∫_{-π/2}^{π/2}cos(cosθx)cos(x)dx = 2cos(cosθ(π/2))/(sin^{2}θ).

E_{θ}(**r**,t) = [I_{0}/(2πε_{0}cr)]cos(ωt-kr)cos[(π/2)cosθ)]/sinθ.

B_{f}(**r**,t) = [μ_{0}I_{0}/(2πr)]cos(ωt-kr)cos[(π/2)cosθ)]/sinθ.

**S**= (1/μ_{0})**E**×**B**= (**r**/r)[I_{0}^{2}/(4π^{2}ε_{0}cr^{2})]cos^{2}(ωt-kr)cos^{2}[(π/2)cosθ)]/sin^{2}θ.

The plot below compares the angular radiation pattern of a dipole (sin^{2}θ) and a half-wave antenna (cos^{2}[(π/2)cosθ)]/sin^{2}θ).