Two pieces of the same wire have the same length. From one piece, a square coil containing a sin

Two pieces of the same wire have the same length. From one...

Key Concepts

Relationship between Wire Length and Coil Area

For coils made from a fixed length of wire, the area of the loop (and thus the magnetic dipole moment) depends on the shape of the coil. Different geometrical forms, such as a square or a circle, enclose different areas even if they use the same total length of wire. This relationship is key to comparing performances of coils of different shapes.

Geometric Properties of Square and Circular Loops

The calculation of the area enclosed by a loop depends on its geometry. For a square, the area can be derived from its perimeter, and for a circle, the area is related to its circumference. Knowing how to derive the areas from the given lengths is critical for establishing equivalence in magnetic moments and, consequently, the matching of the resulting torques.

Magnetic Dipole Moment

The magnetic dipole moment of a current-carrying loop is defined as the product of the current and the area enclosed by the loop (? = I × A). This quantity determines the strength and orientation of the magnetic field produced by the loop, and it is central to analyzing the interaction of the coil with an external magnetic field.

Torque on Current Loops

When a current loop with a magnetic dipole moment is placed in a magnetic field, it experiences a torque given by the vector cross product ? = ? × B. This torque tends to align the dipole moment with the magnetic field. Analyzing torque is essential for understanding the orientation behavior of different coils in a uniform magnetic field.

Transcript

00:01 In this problem, we're asked to compare the currents between two loops of wire that have different shapes, but experience a lot of the same circumstances when it comes to torque caused by a magnetic field.

00:17 So what we're going to do is we are going to analyze what it means for the surface area of each of these coils, because they're made into different shapes.

00:28 And then we're going to compare the expressions for the torque between the two in order to compare the currents.

00:34 So what we're given is we're given two lengths of wire.

00:38 They both have the same length and we'll call it big l.

00:43 Now, it's thrashing into two different coils, but both with just one single turn.

00:49 One of the wires is made into a square.

00:54 So we're going to take that long piece of wire fashion into a square, right? and we'll say that each side is little l.

01:01 Now, that means that to relate the total length of the wire with each side of the square, that means the total length can be expressed as four times every individual side of the square.

01:16 Good to know.

01:17 Now, the cross -sectional area of this one -turn square loop is, of course, going to be l squared.

01:25 Now, up here, we have an expression for little l, one side of the square in terms of big l, the whole perimeter or length of the square.

01:34 And we're going to write that in here.

01:35 So we solve this for l.

01:37 L is going to actually be, oh, let me write this here.

01:40 Little l is going to be the total length of divided by 4.

01:44 So we plug that in here, l over 4, square the whole thing.

01:48 We see that the surface area, the cross -sectional area, sorry, of our square loop is given by the total length of the wire that makes that loop divided by 16.

01:57 Square that length and divided by 16.

02:00 Now, the other piece of wire is made into a circle instead, right? and so we know that the total circumference of the circle must be the same as the length of the wire that made the circle, and that's going to be equal to 2 pi r, relating that to the radius of this circle.

02:18 Of course, then r can be expressed as the length of the circle divided by 2 pi.

02:22 Now, we're going to then want to express the cross -sectional area of our circle for this piece of wire is going to be pi r squared, of course, because that is the cross -sectional area of a circle.

02:35 Plug our new expression for r in here, l over 2 pi, and we'll square it, and we see that the area, cross -sectional area of the circle in terms of the circumference of the circle or the length of our wire is given by l squared over 4 pi...

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