The plane of a 5.0 cm ×8.0 cm rectangular loop of wire is parallel to a 0.19 - T magnetic f

step 1 This is chapter 27 problem number 40. We are given a current carrying rectangular loop, a wire,

The plane of a 5.0 cm ×8.0 cm rectangular loop of wire is...

The plane of a 5.0$\mathrm { cm } \times 8.0 \mathrm { cm }$ rectangular loop of wire is parallel to a $0.19 - \mathrm { T }$ magnetic field. The loop carries a current of 6.2$\mathrm { A }$ (a) What torque acts on the loop? (b) What is the magnetic moment of the loop? (c) What is the maximum torque that can be obtained with the same total length of wire carrying the same current in this magnetic field?

The plane of a 5.0$\mathrm { cm } \times 8.0 \mathrm { cm }$ rectangular loop of wire is parallel to a $0.19 - \mathrm { T }$ magnetic field. The loop carries a current of 6.2$\mathrm { A }$ (a) What torque acts on the loop? (b) What is the
magnetic moment of the loop? (c) What is the maximum torque
that can be obtained with the same total length of wire carrying the
same current in this magnetic field?

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment of a current loop is defined as the product of the current in the loop and its area, along with a direction given by the right-hand rule. It characterizes how the current distribution will interact with an external magnetic field, and serves as a fundamental quantity when determining the torque experienced by the loop in the field.

Torque on a Current Loop

When a current loop is placed in an external magnetic field, it experiences a torque that tends to align its magnetic dipole moment with the field. This torque is given by the vector cross product of the magnetic dipole moment and the magnetic field, indicating that both the magnitude of the dipole moment and the angle between it and the magnetic field are crucial in determining the torque.

Loop Geometry and Area

The area of the current loop is a key geometric factor that contributes directly to the magnetic dipole moment. In the case of a loop with a given shape and fixed wire length, the dimensions define the area enclosed, which in turn affects how strongly the loop interacts with the magnetic field.

Optimization for Maximum Torque

For a fixed total length of wire and current, the configuration that maximizes the area will maximize the magnetic dipole moment and the resulting torque. This involves geometrical optimization, such as using a shape (for example, a square) that encloses the maximum possible area for a given perimeter, thereby achieving the highest possible torque in a magnetic field.

Transcript

00:01 This is chapter 27 problem number 40.

00:03 We are given a current carrying rectangular loop, a wire, embedded in an external magnetic field that is parallel to the surface of this loop.

00:14 And the width is given to us as 5 centimeters.

00:18 The length is 8 centimeters.

00:20 That's called it l.

00:22 And the magnitude of the magnetic field is 0 .19 tesla.

00:27 And the current through this closed loop of wires, 6 .2 amps.

00:36 Now, in part a, we are asked to calculate the torque generated by this external b field.

00:41 So, as you know, our equation is going to be the magnitude of current times the magnitude of magnetic field, times the area of the closed loop, times sign of the angle between the normal vector of the area and the magnetic field.

00:57 So, as you know, normal vector is always perpendicular to the plane.

01:04 So in this case, it's going to be out of the plane towards us, which makes the angle between a and b 90 degrees.

01:11 So we can drop this sign term because sine 90 is going to give us one.

01:16 Simply, in order to calculate the torque, then we do i times b times area.

01:22 As you know, it's a rectangle, so it's going to be the length times width.

01:25 So we have 6 .2 amps as are current, and the magnitude of the magnetic fuel is 0 .19 tesla.

01:37 And length is 8 centimeters, so let's convert it to meters.

01:43 In times we have, i'll go here, times 5 tenths and negative 2, again, meters.

01:49 When we do the calculation, what we find is 0 .4 .7, 1, 2, 5 .5 .5 .5 .5 .5 .5 .5 .2 .2 .5 .2.

01:56 Mutin meters as our torque.

02:00 Now in part b, we are asked to calculate the magnetic moment...

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