A square loop of wire, of side d , carries a current I . Show that the magnetic field at the cen

A square loop of wire, of side d , carries a current I ....

Key Concepts

Biot-Savart Law

The Biot-Savart Law is a fundamental principle used to calculate the magnetic field produced by a current element at a given point in space. It relates the magnetic field to the current element, the distance from the element, and the angle between the element and the line connecting the element to the point, making it essential for integrating contributions from different parts of a current-carrying conductor.

Superposition Principle

The superposition principle states that the total magnetic field created by multiple current elements is the vector sum of the magnetic fields produced by each element individually. This principle is crucial when determining the net magnetic field at a point due to contributions from distinct segments of a current loop or any complex current configuration.

Magnetic Field Due to a Finite Current-Carrying Wire

Calculating the magnetic field from a finite segment of wire involves integrating the Biot-Savart Law over the length of the segment. This method uses the geometrical relationship between the wire and the point of interest, often resulting in expressions that depend on the angles subtended by the ends of the wire at that point.

Symmetry in Magnetic Field Calculations

Symmetry plays a key role in simplifying magnetic field calculations, especially in configurations with regular geometries like loops and polygons. By recognizing symmetric properties, one can reduce the computational effort by evaluating the field contribution from one segment and then extending the result to the entire configuration.

Transcript

00:01 Our question wants us to show that at the center of a loop wire of side d, carrying current i, that the magnetic field is 2, root 2 times mu not i over pi d.

00:13 Okay, so in order to do that, i went ahead and drew out the diagram of what we have here, where we have this loop that goes from point a to b to c to d, carrying current i going in the clockwise direction.

00:25 So if you, using the right -hand rule, if you curl your fingers in the direction of i, the magnetic field then would be into the page.

00:32 So since each side is length d to the center there in the diagram that i drew is a distance d over 2.

00:39 So what we can say here is that the length l to the center is the distance d divided by 2.

00:47 And then you have these two angles that it would make here, angle alpha and beta for just considering one side.

00:52 So we can find the magnetic field from each side of this loop and then add them all together to get the total magnetic field at the center.

01:01 So the angle alpha here that we're considering is going to be equivalent to the angle beta, and these are equal to 45 degrees.

01:13 Okay, so we can use the formula of the magnetic field due to a straight wire of finite length for each side of these squares.

01:20 And then like i said, the field of all the sides is going to add.

01:24 So you'll consider all four sides to calculate the total strength.

01:29 So in general, the magnetic field due to a finite length, well, actually, why don't we write it this way? so the total magnetic field b is going to be the magnetic field of each side, right? b1 plus b2 plus b3 plus b4.

01:52 So now we just need to find the magnetic field due to one side.

01:55 So we'll start with b1.

01:57 B1, the equation here is mu not, the magnetic permeability of free space, times the current, which is i, divided by 4 pi times the distance l, or the length.

02:11 But we said that l was d over 2 here.

02:13 So this is going to be times d over 2.

02:18 And then this gets multiplied by the sign of alpha plus the sine of beta.

02:25 We also know that alpha and beta are equivalent, and they're both equal to 45 degrees.

02:32 So we can further simplify this a little bit.

02:36 And we see that this is equal to, let's actually write it on another page, so we're not pressed for room here.

02:42 B1 is equal to...