# Biot-Savart equation for AC current

• Artemisia_
In summary, 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?
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.

Also, it helps if you can start to learn how to post math equations using LaTeX. There is a "LaTeX Guide" link at the bottom of the Edit window.

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

bob012345
Artemisia_ said:
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
kuruman said:
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
kuruman said:
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.

Delta2 said:
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
berkeman said:
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.

Thank you all for the replies, i try to give here a bit more information, the frequency range is lower than 1KHz, so not extrememly high, and the magnetic field calculation i want to have it in the centre of the coil/along the central axis of the coil, so not in all the space.

Delta2

## What is the Biot-Savart equation for AC current?

The Biot-Savart equation is a fundamental law in electromagnetism that describes the magnetic field produced by a steady electric current. It states that the magnetic field at a certain point is directly proportional to the magnitude of the current, the length of the current-carrying conductor, and the sine of the angle between the direction of the current and the line connecting the point to the conductor.

## How is the Biot-Savart equation used in AC circuits?

In AC circuits, the Biot-Savart equation is used to calculate the magnetic field produced by alternating currents. This is important in understanding the behavior of electrical devices such as motors and transformers, which rely on magnetic fields to function.

## What are the limitations of the Biot-Savart equation for AC current?

The Biot-Savart equation is only applicable to steady currents and does not take into account the changing direction of the current in AC circuits. It also assumes that the current is flowing through a straight conductor, which may not always be the case in practical applications.

## How is the Biot-Savart equation derived?

The Biot-Savart equation is derived from the laws of electromagnetism, specifically Ampere's law and the Lorentz force law. It is a mathematical representation of the relationship between electric currents and magnetic fields.

## What are some real-world applications of the Biot-Savart equation for AC current?

The Biot-Savart equation is used in a variety of fields, including electrical engineering, physics, and geophysics. It is used to design and analyze the performance of electrical devices, as well as to study the Earth's magnetic field and its effects on navigation systems.

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