# Prove that the torque of any current loop is m X B

• Happiness
In summary, the conversation discusses problem 6.2 of Griffith's "Introduction to Electrodynamics" and how to show that the torque on any steady current distribution in a uniform field is equal to the magnetic moment times the magnetic field. The solution involves using the Lorentz force law and a useful identity, and clarifies that the difference between two infinitesimal vectors is the same as the infinitesimal vector itself.
Happiness

## Homework Statement

Problem 6.2 of Griffith's "Introduction to Electrodynamics": Starting from the Lorentz force law ##\vec F=\int I (d\vec l \times \vec B)##, show that the torque on any steady current distribution (not just a square loop) in a uniform field ##\vec B## is ##\vec m\times \vec B##. (##\vec m## is the magnetic moment.)

## Homework Equations

Let the torque be ##\vec N##.

##d\vec N = \vec r\times d\vec F##.

## The Attempt at a Solution

Useful identity: ##\oint \vec r\times d\vec l = 2\vec a##, where ##\vec a## is the area of the loop and points perpendicularly to its surface.

My question: the solution says that ##d\vec r = d\vec l##, which I don't understand. They are clearly pointing in different directions. ##d\vec r## points in the direction from the origin to the point ##r##, while ##d\vec l## points in the direction of the wire of the loop, which in general is different from the direction of ##d\vec r##.

The solution:

It is ##\vec{r}## that points from the origin to the point r =(x,y,z) of the loop and ##\vec{r'}=\hat{i}(x+dx)+\hat{j}(y+dy)+\hat{k}(z+dz)## that points from the origin to the point r'=(x+dx,y+dy,z+dz) of the loop. Their difference is the ifinitesimal vector ##d\vec{r}=\hat{i}dx+\hat{j}dy+\hat{k}dz## which is the same as dl.

Happiness
Delta² said:
It is ##\vec{r}## that points from the origin to the point r =(x,y,z) of the loop and ##\vec{r'}=\hat{i}(x+dx)+\hat{j}(y+dy)+\hat{k}(z+dz)## that points from the origin to the point r'=(x+dx,y+dy,z+dz) of the loop. Their difference is the ifinitesimal vector ##d\vec{r}=\hat{i}dx+\hat{j}dy+\hat{k}dz## which is the same as dl.

Thanks a lot! I've got it now.

## 1. What is torque and how is it related to current loops?

Torque is a measure of the rotational force on an object. In the case of a current loop, torque is the force that causes the loop to rotate when placed in a magnetic field. This torque is directly proportional to the strength of the magnetic field and the current in the loop.

## 2. How is the torque of a current loop calculated?

The torque of a current loop is calculated by multiplying the magnetic field strength (B) by the current (I) and the area of the loop (A). This can be represented by the equation: τ = BIA. The direction of the torque is determined by the direction of the magnetic field and the direction of the current in the loop.

## 3. What is the significance of the cross product in the equation for torque?

The cross product in the equation for torque represents the vector nature of torque. This means that the torque is not just a simple multiplication of the values, but also takes into account the direction of the magnetic field, the current, and the area of the loop. It also shows that torque is a rotational force, rather than a linear force.

## 4. Can you provide an example of a current loop experiencing torque in a magnetic field?

One example of a current loop experiencing torque in a magnetic field is a simple DC motor. The current-carrying loop, or coil, is placed in a magnetic field and when the current flows through the coil, it experiences a torque that causes it to rotate. This is how the motor converts electrical energy into mechanical energy.

## 5. How does the direction of the magnetic field and current affect the direction of the torque?

The direction of the torque is determined by the right-hand rule, which states that if the thumb of the right hand points in the direction of the current, and the fingers point in the direction of the magnetic field, then the palm of the hand points in the direction of the torque. This means that the direction of the torque can be changed by either changing the direction of the current or the direction of the magnetic field.

Replies
1
Views
799
Replies
13
Views
1K
Replies
1
Views
1K
Replies
4
Views
2K
Replies
11
Views
1K
• Introductory Physics Homework Help
Replies
25
Views
527
Replies
1
Views
2K
Replies
1
Views
1K
• Introductory Physics Homework Help
Replies
12
Views
522
• Introductory Physics Homework Help
Replies
1
Views
352