# Biot-Savart equation for AC current

• Artemisia_

#### Artemisia_

Homework Statement
Considering to have a uniform sinusoidal current (in time) flowing in a circular loop, how does the sinusoidal nature of the current affects the magnetic field generated by the coil?
Relevant Equations
B(r) = (m0/4pi)integ(i*(dl*(x-l)/(x-l)^3))
The fact that the current changes in time should allow me to take it outside from the integral along the coil, since it is not a function of space. But I'm not sure about this assumption, is there something I'm missing or am i correct?
Thanks

Welcome to PF.

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Biot-Savart is applicable at all times. If the current is time-dependent, you replace ##i## in the expression with ##i(t)##.

• bob012345
Homework Statement:: Considering to have a uniform sinusoidal current (in time) flowing in a circular loop, how does the sinusoidal nature of the current affects the magnetic field generated by the coil?
Relevant Equations:: B(r) = (m0/4pi)integ(i*(dl*(x-l)/(x-l)^3))

The fact that the current changes in time should allow me to take it outside from the integral along the coil, since it is not a function of space. But I'm not sure about this assumption, is there something I'm missing or am i correct?
Thanks
Biot-Savart is applicable at all times. If the current is time-dependent, you replace ##i## in the expression with ##i(t)##.
And there is the additional consideration of whether this is a circular loop antenna, radiating EM away from the near field equations to the far field EM propagation equations.

• vanhees71 and Delta2
Biot-Savart is applicable at all times. If the current is time-dependent, you replace ##i## in the expression with ##i(t)##.
I guess you mean this in the quasi static approximation (when we neglect the displacement current term in Maxwell's equations). Because in the full dynamic treatment a time varying current will create time varying magnetic field, which will create time varying electric field, which according to the displacement current term in the Maxwell's- Ampere's law will create additional magnetic field.
Biot-Savart law is equivalent to Ampere's law without the displacement current term.

I guess you mean this in the quasi static approximation (when we neglect the displacement current term in Maxwell's equations). Because in the full dynamic treatment a time varying current will create time varying magnetic field, which will create time varying electric field, which according to the displacement current term in the Maxwell's- Ampere's law will create additional magnetic field.
Biot-Savart law is equivalent to Ampere's law without the displacement current term.
Sure. This is an introductory level question and that is why I did not consider the displacement current. Nevertheless, we have to hear again from OP as @berkeman suggested.

• Delta2 and berkeman
I think there is another reason why we should consider the quasi static approximation, the current in the loop is given as uniform, therefore according to an approximation we can make (can't reveal more since we wait to hear from OP) the far field (##\frac{1}{r}## term) is zero.

• kuruman
And there is the additional consideration of whether this is a circular loop antenna, radiating EM away from the near field equations to the far field EM propagation equations.

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