Torque On Current Loop And Magnetic Moment Derivation - Practice Questions & MCQ

Torque on a rectangular current loop in a uniform magnetic field - 

As we have studied that the electric dipole in a uniform electric field it will experience a torque similarly if we place a rectangular loop carrying a steady current and placed in a uniform magnetic field experiences a torque. It does not experience a net force.

Let us consider a case when the rectangular loop is placed such that the uniform magnetic field B is in the plane of the loop. This is illustrated in the given figure.The field exerts no force on the two arms AD and BC of the loop. It is perpendicular to the arm AB of the loop and exerts a force F₁ on it which is directed into the plane of the loop. Its magnitude is,

                                                                                              $F_1=I b B$

Similarly it exerts a force F₂ on the arm CD and F₂ is directed out of the plane of the paper.

                                                                                        $F_2=I b B$

Thus, the net force on the loop is zero. But these two forces are actingat a distance 'a' between them. This torque on the loop due to the pair of forces F₁ and F₂ . From the figure given below shows that the torque on the loop tends to rotate it anti-clockwise. This torque is (in magnitude),

$
\begin{aligned}
\tau & =F_1 \frac{a}{2}+F_2 \frac{a}{2} \\
& =I b B \frac{a}{2}+I b B \frac{a}{2}=I(a b) B \\
& =I A B
\end{aligned}
$

where $A=a b$ is the area of the rectangle.
Now we will discuss the case when the plane of the loop is making an angle $\theta$ with magnetic field. In the previous case we have considered $\theta=\frac{\pi}{2}$, but this is now a general case.

Here again you can see that the forces on arms $A B$ and $C D$ are $F_1$ and $F_2$ -

$
F_1=F_2=I b B
$

Then the torque will be the

$
\begin{aligned}
\tau & =F_1 \frac{a}{2} \sin \theta+F_2 \frac{a}{2} \sin \theta \\
& =\mathrm{Iab} B \sin \theta \\
& =I A B \sin \theta
\end{aligned}
$
 

From the above equations we can see that the torques can be expressed as vector product of the magnetic moment of the coil and the magnetic field. We define the magnetic moment of the current loop as,

$
m=I A
$

If the coil has N turns then the magnetic moment formula becomes -

$
m=N I A
$

And its direction is defined by the direction of Area vector.
So, Torque equation can be written as,

$
\tau=\mathbf{m} \times \mathbf{B}
$
 

Circular current loop as magnetic dipole-

From the previous concept, we have studied the magnetic field due to a current $(\mathrm{l})$ carrying circular wire (Radius $=\mathrm{R}$ ) on its axis at a distance ' x ' is -

$
B=\frac{\mu_0 I R^2}{2\left(x^2+R^2\right)^{3 / 2}}
$

If $x>>>R$, then R will become negligible and the equation becomes -

$
B=\frac{\mu_0 I R^2}{2 x^3}
$

Now, as the area of this loop is $\pi R^2$, so the equation becomes -

$
B=\frac{\mu_0 I A}{2 \pi x^3}
$

As, $m=I A S o$,

$
\begin{aligned}
\mathbf{B} & =\frac{\mu_0 \mathbf{m}}{2 \pi x^3} \\
& =\frac{\mu_0}{4 \pi} \frac{2 \mathbf{m}}{x^3}
\end{aligned}
$
 

The expression shown above is very similar to an expression obtained earlier for the electric field of a dipole.
The similarity may be seen if we substitute,

$
\begin{aligned}
& \mu_0 \rightarrow 1 / \varepsilon_0 \\
& \mathbf{m} \rightarrow \mathbf{p}_e \text { (electrostatic dipole) } \\
& \mathbf{B} \rightarrow \mathbf{E} \text { (electrostatic field) }
\end{aligned}
$

So the equation become,

$
\mathbf{E}=\frac{2 \mathbf{p}_e}{4 \pi \varepsilon_0 x^3}
$

So, We can say from the above analogy that the circular current loop can act as a magnetic dipole. The direction of the magnetic moment can be obtained as -

So, We can say from the above analogy that the circular current loop can act as a magnetic dipole. The direction of the magnetic moment can be obtained as - 

But there is a fundamental difference: an electric dipole is built up of two elementary units — the charges (or electric monopoles). In magnetism, a magnetic dipole (or a current loop) is the most elementary element. The equivalent of electric charges, i.e., magnetic monopoles, are not known to exist.