Biot-Savart non-textbook Equation -- B at point above a loop but off-axis

In summary, the magnetic field around a closed loop can be calculated by taking the derivative of the field with respect to position.
  • #1
I read everywhere about the formulas for calculating B at a point from a length of straight wire, or at a point from the centre of a closed loop.
But what about at a point over a closed loop that wasn't the centre? Is there a simple calculation for that?

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  • #3
Thanks for the reply zoki, I only had a quick look, but didn't those examples still only deal with a point still being on the center (z) axis of the loop? (not like directly over the loop or closer to one side than the other)
I'll have a better look later, in which case if I'm wrong I'll remove this post.
  • #4
Attached is a plot of the field of a loop. This graph shows the magnetic field you would expect to measure as you walk across a wire loop resting on the ground surface. The loop is shown as the circle, viewed from above, modeled with many short segments as shown. The horizontal scale is in radii.
The plot drawn across the graph is the field that is seen traveling along a line that passes through the axis, parallel to, but slightly away from the plane of the loop. The plot is normalised so that +Bo is the axial field value.
The steepness of the transition while crossing the wire is determined by the offset from the plane of the coil. Let me know if you find an analytic solution.
B_Field Circular _Loop .png
  • #5
I'm not sure I get it completely (cool graph by the way), I'm really sorry.
So there is a wire loop of current on the ground, I'm walking above it from -2 to +2, passing through the center of the axis of symmetry. I don't see how the yellow graph can be B? All I can picture is current traveling, say, anti-clockwise, then I would have through if the yellow graph was B, that as I approach it and am at -1 it peaks and all the B is coming from the LHS then as I am at the middle, the only component is from the center and there is some cancellation diagonally, then as I leave a gain at 1 it peaks with just RHS magnetic field. So why would it go negative as I first approach it, and leave why would it have a negative gradient? I would have thought it would taper up as I approach, peak, taper down (due to cancellation), taper up, peak, taper down.

Thanks heaps!
  • #6
So I thought it would look like this:
  • #7
Sorry, the graph is the vertical component of the field.

The vertical field inside the loop is opposite that outside the loop.

Over the wire the B field is horizontal so it has no vertical component.
  • #8
Ok that makes sense, but what if you wanted the total B strength, not just a component, over the path from left to right?
Would you sum the components, dot product, or cross product?
  • #9
You would compute Bhorizontal and Bvertical, the total is the vector sum, B = √(Bh2 + Bv 2)
  • #10
Ah of course,
Thank you very much

What is the Biot-Savart non-textbook equation?

The Biot-Savart non-textbook equation is a mathematical formula used to calculate the magnetic field (B) at a point located above a current-carrying loop, but off-axis. It takes into account the magnitude and direction of the current, as well as the distance and angle from the point to the loop.

Why is this equation important in science?

This equation is important because it allows scientists to predict and calculate the magnetic field at any given point around a current-carrying loop. This information is crucial in understanding and studying electromagnetism and its applications in various fields, such as physics, engineering, and biology.

What are the variables in the Biot-Savart non-textbook equation?

The variables in this equation are the permeability of free space (μ0), the current (I), the distance from the point to the loop (r), the angle between the current and the position vector (θ), and the length of the loop (L).

How is the Biot-Savart non-textbook equation derived?

This equation is derived from the Biot-Savart law, which states that the magnetic field at a point is directly proportional to the current and inversely proportional to the distance from the point to the current element. The non-textbook form of the equation takes into account the off-axis position of the point from the current-carrying loop.

Are there any limitations to this equation?

Like any mathematical formula, there are limitations to the Biot-Savart non-textbook equation. It assumes that the current is uniformly distributed along the loop and that the loop is infinitely thin. It also does not take into account the effects of other nearby currents or magnetic fields. Additionally, it is only applicable to steady-state currents and does not account for time-varying fields.

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