A Hertzian dipole is located at the origin of spherical coordinates and is align
ID: 1894379 • Letter: A
Question
A Hertzian dipole is located at the origin of spherical coordinates and is aligned with the theta=0 direction. The dipole has strength I(subscript 0) delta l and oscillates with angular frequency omega. The magnetic field that it produces is given by the real part of the expression
B(r)=[((mu(subscript 0) I(subscript 0) delta l))/4pi] sin theta ((-i omega/rc)+(1/(r^2)) exp[i(kr - omega t)] phi-hat
When grouping together terms by r-dependence, there are essentially three contributions to the fields produced by a Hertzian dipole: the r^-3 terms are effectively the electrostatic field; the r^-2 terms give rise to what is called the induction field; and, the radiation field is the single term in r^-1.
Show that, in the limit of small distances r and zero angular frequency, the amplitude of the field given by this expression is consistent with the Biot-Savart law:
B(r)=[(mu(subscript 0) I)/4pi][(delta l cross r-hat)/(r^2)]
Explanation / Answer
B(r)=[(µ(subscript 0) I(subscript 0) d l)/(4pi)] sin? [((1/(r^2))]exp [i(kr-?t)] phi-hat B(r)=[(µ(subscript 0) I(subscript 0) d l)/(4pi)] sin? [((1/(r^2))] cos(kr-?t) phi-hat B(r)=[(µ(subscript 0) I(subscript 0) d l)/(4pi)] sin? [((1/(r^2))] cos(kr) phi-hat B(r)=[(µ(subscript 0) I(subscript 0) d l)/(4pi)] sin? [((1/(r^2))] cos(0) phi-hat because r is small B(r)=[(µ(subscript 0) I(subscript 0) d l)/(4pi)] sin? [((1/(r^2))] phi-hat I suppose I need to find out what theta would be, but I am having trouble visualising the thing in my head. I suppose theta would have to be 90 degrees hope this helps
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