Why Does the Toroid's Magnetic Field Equation Use a Cubed Distance Term?

[richyw]

Homework Statement

I'm working through a proof that says the magnetic field of a toroid is circumferential at all points inside and outside the toroid. I can follow most of the proof, but am a bit confused where the first equation comes from.

Here is the figure from the textbook (Griffith's 4th Ed).

http://media.newschoolers.com/uploads/images/17/00/67/76/17/677617.jpeg

To begin the proof, Griffiths starts with the field at $\mathbf{r}$ due to the current element at $\mathbf{r}'$.$$d\mathbf{B}=\frac{\mu_0}{4\pi}\frac{\mathbf{I}\times\mathbf{\hat{{u}}}}{u^3}dl'$$

Homework Equations

$$\mathbf{B}=\frac{\mu_0}{4 \pi}\int\frac{\mathbf{I}\times\mathbf{\hat{u}}}{u^2}dl'$$

The Attempt at a Solution

I'm just confused at how Griffiths got from the Biot-Savart law above, into the equation he posted in the question. (I replaced the script r with a u). Where does the cubed term come from?

 

[xophergrunge]

I believe that is a mistake in the text. I think he meant it to be the vector r and not r hat, in which case the denominator would be cubed.

 

[rude man]

I agree with post #2. Otherwise the dimensions don't check out.

 

[richyw]

cool thanks. I was thinking that, but I wanted to confirm!

 

[HalfLostAndSearching]

I would like to clarify that the first equation in the question is not the Biot-Savart law, but rather the magnetic field due to a current element. The Biot-Savart law is a general formula for calculating the magnetic field due to a current-carrying wire or a current element.

To answer the question about where the cubed term comes from, it is a result of the vector cross product in the numerator of the equation. The magnitude of the magnetic field depends on the magnitude of the current, the distance between the current element and the point of interest, and the angle between the current element and the direction of the magnetic field. This is represented by the magnitude of the vector cross product, which is proportional to the sine of the angle between the two vectors. When we take the cross product of the current element and the unit vector in the direction of the magnetic field, we get a vector with a magnitude of sin(theta). This vector is then divided by the distance cubed (u^3) to account for the inverse square law of the magnetic field.

I hope this explanation helps clarify the origin of the first equation and how it relates to the Biot-Savart law.