\( dx^{'}\quad =\quad 2\pi dr\)
\( B=-\cfrac { dE }{ dx^{'} } =-\cfrac { 1 }{ 2\pi } \cfrac { dE }{ dr } \) ----(*)
Consider a wire carrying current, \(I\).
\(B=\cfrac { I }{ 2\pi \varepsilon _{ o }r } \cfrac { 1 }{ v } \)
where \(r\) is the distance perpendicular to the wire.
\(V=IR\), a wire of length \(d\), \(I=\cfrac{E.d}{R}\)
\( B=\cfrac { d }{ Rv } .\cfrac { 1 }{ 2\pi \varepsilon _{ o }r } E\)
Equating to (*)
\( \cfrac { d }{ Rv } .\cfrac { 1 }{ 2\pi \varepsilon _{ o }r } E =-\cfrac { 1 }{ 2\pi } \cfrac { dE }{ dr } \)
\( -\cfrac { d }{ R\varepsilon _{ o }v } \int {\cfrac { 1 }{ r }} dr=\int{\cfrac { 1 }{ E }} dE\)
\( -\cfrac { d }{ R\varepsilon _{ o }v } ln(r)=ln(E)\)
\( E=C{ r }^{ -\cfrac { d }{ R\varepsilon _{ o }v } }\)
Since \(E=0\) as \(r\rightarrow\infty\) the only reasonable solution is \(C=0\). Which means \(E\) is zero for all \(r\) through the length of the conductor. This is consistent with the fact that the E field outside of a current carrying wire is zero.
But from, \( B=-\cfrac { dE }{ dx^{'} }\) we are able to derive,
\( \cfrac { \partial ^{ 2 }E }{ \partial t^{ 2 } } =\cfrac { \partial ^{ 2 }E }{ \partial x^{ 2 } } v^{ 2 }\)
That is to say, when a potential function \(V=d.E\) is applied to the wire, that \(E\) satisfies the wave equation above then we have a wave propagating in the \(x\) direction parallel to the length of the wire. Combining this with the fact that E field outside of the conductor along its length is zero,
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| Radiation Pattern of a Wire Antenna |
one possibility is that the wave is emanating from the end of the wire in semi-spherical wave front and is zero below that point. The intensity of the wave decreases by a factor of \(\cfrac{1}{2}*4\pi r^2=2\pi.r^2\) where \(r\) is the radial distance from the end of the wire. Another wire slightly lower than the transmitting wire antenna will not be affect by it, reflected signals from the ionosphere aside. The receiving wire must be higher that the transmitting wire to receive from it, especially if the signal is too weak to bounce off the ionoshpere. A reflecting back plane at the transmitting wire is useless because the wave does not reach behind the end point. The wire is not behaving like a point source. The intensity of the wave from the end of the wire is twice that from a point source.
So, if you want to receive from a straight antenna directly you have to be higher than the antenna, otherwise point your receiving antenna upwards to receive reflected signals from the ionosphere or at some tall building higher than the antenna, reflecting signals from the antenna.
