A square coil and a rectangular coil are each made from the...

A square coil and a rectangular coil are each made from the same length of wire. Each contains a single turn. The long sides of the rectangle are twice as long as the short sides. Find the ratio $\tau_{\text { square }} / \tau_{\text { rectangle }}$ of the maximum torques that these coils experience in the same magnetic field when they contain the same current.

A square coil and a rectangular coil are each made from the same length of wire. Each contains a single
turn. The long sides of the rectangle are twice as long as the short sides. Find the ratio $\tau_{\text { square }} / \tau_{\text { rectangle }}$ of the maximum torques that these coils experience in the same magnetic field when they contain the same current.

Key Concepts

Coil Area Determination from a Fixed Perimeter

When a coil is made from a fixed length of wire, its area is determined by its shape and the way this length is distributed among its sides (i.e., the perimeter constraint). For a square coil, the area is obtained by dividing the total perimeter equally among four sides. For a rectangle with a given ratio of side lengths, the area must be calculated using the specified proportional dimensions, making it critical to account for these constraints when computing the effective area.

Relating Coil Area to Torque Ratios

Since the maximum torque experienced by a coil in a magnetic field is directly proportional to its area (assuming identical current and field strength), the ratio of the torques for different coil shapes can be determined by computing the ratio of their respective areas. This concept allows one to analyze how variations in geometry affect the performance of coils in magnetic fields.

Magnetic Torque on a Current Loop

A current-carrying coil in a magnetic field experiences a torque that tends to align the plane of the coil with the magnetic field. This maximum torque is given by the product of the current, the magnetic field strength, and the area of the coil when the field is perpendicular to the normal of the coil. Understanding this relationship is essential for comparing the torques of different coil shapes.

Transcript

00:02 Okay, so our objective for this problem is to find a ratio of torques for two wires in a magnetic field.

00:12 So we have our equation for torque is number of coils times the current, times the area, times the magnetic field, sine of phi.

00:28 And we have two different setups.

00:30 We have a square and a rectangle.

00:32 And we want the ratio of the square to the rectangle.

00:38 And we want to find when the torque is maximum, which in this case is when the sign of phi is equal to 1, which we have when phi is equal to 90 degrees.

00:50 90 degrees is when our magnetic field is straight perpendicular to our area enclosed by the wire.

00:59 So when phi is 90 degrees, we have our maximum torque.

01:04 And our two setups, we have a square, a wire that is in a perfect square.

01:09 And we have a wire that is more rectangular shaped.

01:13 So we want to find the ratios of torque.

01:18 So we have everything's going to be the same in both cases other than the area.

01:22 So what will end up having all of this over the same thing, except we'll have a different area.

01:30 I'll call that area too here.

01:36 That contributes to the torque.

01:38 Everything else will cancel out.

01:40 Our magnetic fields will cancel, our currents will cancel, and our number of coils will cancel.

01:48 So all we have to worry about are the two areas, and we'll get the same ratio.

01:53 So what we want is the ratio of the area for the square divided by the area for the rectangle.

02:04 So let's see if we can relate these two.

02:09 So we know that our total length is split up into four for each side of the square.

02:15 We'll call our length l.

02:17 So each one is a quarter, so l over 4, which means our area is one side times the other, which would give you l squared times 4 squared.

02:29 And we have for our rectangle that one side is twice the other side...

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