Question 9: Magnetism and Matter

Question

An insulated wire is wound so that it forms a flat coil with $N=200$ turns. The radius of the innermost turn is $r_1=3 \mathrm{~cm}$, and of the outermost turn $r_2=6 \mathrm{~cm}$. If 20 mA current flows in it then the magnetic moment will be $\alpha 	imes 10^{-2} \mathrm{~A} . \mathrm{m}^2$. The value of $\alpha$ is $\_\_\_\_$ .

Accepted Answer

Given:
Charge, $q = 10^{-9}\,	ext{C}$
Electric field, $\vec E = 0.4\,\hat j\ 	ext{N/C}$
Magnetic field, $\vec B = 4	imes 10^{-3}\,\hat k\ 	ext{T}$
Force, $\vec F = (4\hat i + 2\hat j)	imes 10^{-10}\ 	ext{N}$
We use the Lorentz force formula:
$$ \vec F = q\left(\vec E + \vec v 	imes \vec Bight) $$
Since the particle is moving in the $x$-$y$ plane, let
$$ \vec v = v_x \hat i + v_y \hat j $$
Now calculate $\vec v 	imes \vec B$:
$$ \vec v 	imes \vec B = (v_x \hat i + v_y \hat j)	imes \left(4	imes 10^{-3}\hat kight) $$
Using cross products:
$$ \hat i 	imes \hat k = -\hat j,\qquad \hat j 	imes \hat k = \hat i $$
So,
$$ \vec v 	imes \vec B = 4	imes 10^{-3}\left(v_y \hat i - v_x \hat jight) $$
Now,
$$ \vec E + \vec v 	imes \vec B = 0.4\hat j + 4	imes 10^{-3}v_y \hat i - 4	imes 10^{-3}v_x \hat j $$
$$ = \left(4	imes 10^{-3}v_yight)\hat i + \left(0.4 - 4	imes 10^{-3}v_xight)\hat j $$
Multiplying by $q = 10^{-9}$:
$$ \vec F = 10^{-9}\left[\left(4	imes 10^{-3}v_yight)\hat i + \left(0.4 - 4	imes 10^{-3}v_xight)\hat jight] $$
This is equal to the given force:
$$ (4\hat i + 2\hat j)	imes 10^{-10} $$
Now compare components.
For $\hat i$ component:
$$ 10^{-9}\left(4	imes 10^{-3}v_yight) = 4	imes 10^{-10} $$
$$ 4	imes 10^{-12}v_y = 4	imes 10^{-10} $$
$$ v_y = 100\ 	ext{m/s} $$
For $\hat j$ component:
$$ 10^{-9}\left(0.4 - 4	imes 10^{-3}v_xight) = 2	imes 10^{-10} $$
$$ 0.4 - 4	imes 10^{-3}v_x = 0.2 $$
$$ 4	imes 10^{-3}v_x = 0.2 $$
$$ v_x = 50\ 	ext{m/s} $$
Hence,
$$ \vec v = 50\hat i + 100\hat j\ 	ext{m/s} $$
So, the correct option is:
$$ \boxed{50\hat i + 100\hat j} $$
Option A

Answer

4.4

Answer

2.64

Answer

3.25

Answer

1.2

Magnetism and Matter

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