In Fig. 28-47, a rectangular loop carrying current lies in...

In Fig. $28-47,$ a rectangular loop carrying current lies in the plane of a uniform magnetic field of magnitude 0.040 T. The loop consists of a single turn of flexible conducting wire that is wrapped around a flexible mount such that the dimensions of the rectangle can be changed. (The total length of the wire is not changed.) As edge length $x$ is varied from approximately zero to its maximum value of approximately $4.0 \mathrm{cm},$ the magnitude $\tau$ of the torque on the loop changes The maximum value of $\tau$ is $4.80 \times$ $10^{-8} \mathrm{N} \cdot \mathrm{m} .$ What is the current in the loop?

In Fig. $28-47,$ a rectangular loop carrying current lies in the plane of a uniform
magnetic field of magnitude 0.040 T. The
loop consists of a single turn of flexible conducting wire that is wrapped around a flexible mount such that the dimensions of the rectangle can be changed. (The total length
of the wire is not changed.) As edge length $x$ is varied from approximately zero to its
maximum value of approximately $4.0 \mathrm{cm},$ the magnitude $\tau$ of
the torque on the loop changes The maximum value of $\tau$ is $4.80 \times$
$10^{-8} \mathrm{N} \cdot \mathrm{m} .$ What is the current in the loop?

Key Concepts

Magnetic Dipole Moment

The magnetic dipole moment of a current loop represents a measure of the loop’s tendency to align with a magnetic field. It is defined as the product of the current flowing through the loop and the area enclosed by that loop (m = I × A). This concept is fundamental in understanding how loops interact with magnetic fields, as the magnitude and direction of the dipole moment determine the strength of the interaction.

Torque in a Magnetic Field

A current loop in a uniform magnetic field experiences a torque due to the interaction between its magnetic dipole moment and the external field. This torque is given by the relation ? = mB sin?, where m is the magnetic dipole moment, B is the magnetic field strength, and ? is the angle between the dipole moment and the field direction. Maximum torque occurs when sin? is equal to one, meaning the magnetic moment is perpendicular to the field.

Optimization of Area for a Given Perimeter

When the total length of the wire forming a current loop is fixed, the shape of the loop can be adjusted so that the area enclosed by it is maximized. In the case of a rectangle, the maximum area for a fixed perimeter is obtained when the rectangle is a square. This optimization is critical in problems where increasing the area (and thus the dipole moment) under a constraint directly affects the magnitude of forces or torques experienced by the loop.

Transcript

00:01 So here is a rectangular loop which shape can be changed like from x can go from 0 like this this is where x equal to 0 2 like this where x equals to max x equal is equal to 4 right so its length can this edge can be changed this edge can be changed when from x equal to 0 to x equal to 4 so when the length was changed from x equal to x equal to 4 then there was a maximum torque at us at some point of at some value of x t max is equal to 4 .8 into 10 to the power minus 8 nrometer so what we have to find out it what is what is the current in the loop? it says that the current the loop is in a magnetic field be of magnitude 0 .0 4 tesla all right so now we have to let us write the formula for torque is equal to magnetic moment magnetical moment cross magnetic field right so magnetic diaper moment is what number of terms into current into area of the wire now since we need t max so for t max mu and b should be perpendicular so that will only give me we don't it's it is sign 90 here so sign 90 is one right for maximum torque t max again the other thing that makes t max is area right so area also should be maximum for t max a also should be maximum so what is the maximum area this and this wire can make so the maximum area for a parallelogram like a rectangle is a square right so what is the so first of all let's find the length of the rectangle the total length the total length l is equal to just look at this if you look at this here so this tends that the total length is 8c right so because the maximum it can you know it can go this edge can go is 4 cm so the maximum length of the wire is 8cm so the total wire is length is 8cm so when it becomes a square then the length of each edge small l is equal to 2cm so this is the case 2 this is the case when the area is maximum and hence the tot is maximum and in this case area equals to a maximum equals to 4cm square right two is a four l square so this is equal to four into ten to the power minus four meters square so now let's continue with our solution here so t max equals to number of tons is one so just let's just write i a max now we are asked to find out i so i equals to t max form a max times b so what is the maximum torque here? that is 4 .8 into the power minus 8.

04:19 What is the maximum area? 4 into 10 to the power minus 4.

04:24 What is 9 .03? 0 .04...

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