Magnetic Effects of Current

Practice Questions:

Question 1: difficult

A long straight wire along the z-axis carries a current I in the negative z-direction. The magnetic
vector field \[\overrightarrow{B}\] at a point having coordinates (x, y) in the z = 0 plane is :

1. \[\frac{\mu_{0}I}{2\pi}\left( \frac{y\hat{i}-x\hat{j}}{x^{2}+y^{2}} \right)\]

2. \[\frac{\mu_{0}I}{2\pi}\left( \frac{x\hat{i}+y\hat{j}}{x^{2}+y^{2}} \right)\]

3. \[\frac{\mu_{0}I}{2\pi}\left( \frac{x\hat{j}-y\hat{i}}{x^{2}+y^{2}} \right)\]

4. \[\frac{\mu_{0}I}{2\pi}\left( \frac{x\hat{i}-y\hat{j}}{x^{2}+y^{2}} \right)\]

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Question 2: difficult

A coaxial cable having radius “a” of inner wire and inner and outer radii “b” and “c” respectively
of the outer shell carries equal and opposite currents of magnitude i on the inner and outer
conductors as shown. What is the magnitude of the magnetic induction at point P of the cable at
a distance r (b < r < c) from the axis?

1. Zero

2. \[\frac{\mu_{0}ir}{2\pi a^{2}}\]

3. \[\frac{\mu_{0}i}{2\pi r}\]

4. \[\frac{\mu_{0}i}{2\pi r}\frac{c^{2}-r^{2}}{c^{2}-b^{2}}\]

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To calculate the magnetic field

$B$

at a point

$P$

within the outer shell of a coaxial cable (

$b < r < c$

), carrying equal and opposite currents

$i$

on the inner and outer conductors, we use Ampère's law and superposition principles.

1. Current Distribution in the Outer Shell

The outer shell carries current

$-i$

, distributed uniformly across the cross-sectional area of the shell between radii

$b$

and

$c$

The current density

$J$

in the shell is:

$J = \frac{-i}{\pi (c^2 - b^2)}.$

The current enclosed within a radius

$r$

(

$b < r < c$

) in the outer shell is the current

$-i$

contributed by the region from

$b$

$r$

$I_{\text{enc,shell}} = J \cdot \text{area of shell portion} = J \cdot \pi (r^2 - b^2).$

Substituting

$J$

$I_{\text{enc,shell}} = \frac{-i}{\pi (c^2 - b^2)} \cdot \pi (r^2 - b^2) = \frac{-i (r^2 - b^2)}{c^2 - b^2}.$

2. Net Enclosed Current at Radius $r$

At any point

$P$

within the shell (

$b < r < c$

), the net current enclosed by a loop of radius

$r$

is:

$I_{\text{enc}} = \text{current from the inner wire} + \text{current from the outer shell}.$

The inner wire contributes

$+i$

, and the shell contributes

$I_{\text{enc,shell}}$

$I_{\text{enc}} = i + \frac{-i (r^2 - b^2)}{c^2 - b^2}.$

Simplify:

$I_{\text{enc}} = i \left(1 - \frac{r^2 - b^2}{c^2 - b^2}\right).$

Factorize:

$I_{\text{enc}} = i \cdot \frac{c^2 - r^2}{c^2 - b^2}.$

3. Magnetic Field at Radius $r$

Using Ampère's law, the magnetic field

$B$

at radius

$r$

is:

$B \cdot 2\pi r = \mu_0 I_{\text{enc}}.$

Substitute

$I_{\text{enc}}$

$B \cdot 2\pi r = \mu_0 \cdot i \cdot \frac{c^2 - r^2}{c^2 - b^2}.$

Solve for

$B$

$B = \frac{\mu_0 i}{2\pi r} \cdot \frac{c^2 - r^2}{c^2 - b^2}.$

Final Answer:

$\boxed{B = \frac{\mu_0 i}{2\pi r} \cdot \frac{c^2 - r^2}{c^2 - b^2}}$

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Question 3: difficult

An electron \left(mass = 9.1\times 10^{-31}kg; Charge=-1.6\times 10^{-19}C \right) experiences no deflection if subjected to an electric field of \[3.2\times 10^{5} V/m\] and a magnetic field of 2.0 × 10-³ Wb/m² . Both the fields are normal to the path of electron and to each other . If the electric field is removed, then the electron will revolve in an orbit of radius:

1. 45 m

2. 4.5 m

3. 0.45 m

4. 0.045 m

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Question 4: difficult

A long wire carrying a current of 2 A is laid along the x axis (current flows along positive
x direction) and another wire carrying current of 4 A is laid along y axis(current flows along
positive y direction). The points at which magnetic field is zero are:

1. (2, 1, 0)

2. (1, 2, 0)

3. (6, 3, 1)

4. (–2, 4, 0)

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Question 5: difficult

Three rings, each having equal radius R, are placed mutually perpendicular to each other
and each having its centre at the origin of coordinate system. If current I is flowing thriugh
each ring then the magnitude of the magnetic field at the common centre is

1. \[\sqrt{3}\frac{\mu_{0}I}{2R}\]

2. Zero

3. \[\left( \sqrt{2}-1 \right)\frac{\mu_{0}I}{2R}\]

4. \[\left( \sqrt{3}-\sqrt{2} \right)\frac{\mu_{0}I}{2R}\]

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Question 6: difficult

In the given figure the magnitude of magnetic field at O is (all three wires are quarter circular arc)

1. \[\frac{\mu_{0}I}{4R}\sqrt{3}\]

2. \[\frac{\mu_{0}I}{2R}\sqrt{3}\]

3. \[\frac{\mu_{0}I}{8R}\sqrt{3}\]

4. none of these

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Question 7: difficult

A current I flows around a closed path in the horizontal plane of the circle as shown in the figure. The path consists of eight arcs with alternating radii r and 2r. Each segment of arc subtends equal angle at the common centre P. The magnetic field produced by current path at point P is

1. \[\frac{3}{8}\frac{\mu_{0}I}{R}\] ,perpendicular to the plane of the paper and directed inward.

2. \[\frac{3}{8}\frac{\mu_{0}I}{R}\] , perpendicular to the plane of the paper and directed outward.

3. \[\frac{1}{8}\frac{\mu_{0}I}{R}\] , perpendicular to the plane of the paper and directed inward.

4. \[\frac{1}{8}\frac{\mu_{0}I}{R}\] , perpendicular to the plane of the paper and directed outward.

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Question 8: difficult

A plastic disc of radius R has a charge q uniformly distributed over its surface. If the disc is rotated with a frequency f about its axis. then magnetic dipole moment will be :

1. \[\frac{\pi qfR^{2}}{2}\]

2. πqfR²

3. 2πqfR²

4. 4πqfR²

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Question 9: difficult

A cube made of wires of equal length is connected to a battery as shown in the figure. The magnetic field at the centre of the cube is :

1. \[\frac{12\mu_{0}I}{\sqrt{2}\pi L}\]

2. \[\frac{6\mu_{0}I}{\sqrt{2}\pi L}\]

3. \[\frac{6\mu_{0}I}{\pi L}\]

4. zero

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Question 10: difficult

Adjoining figure shows a rectangular loop of conductor carrying a current i. The length and breadth of the loop are respectively a and b. The magnetic field at the centre of loop is :

1. \[\frac{\mu_{0}i\left( a+b \right)}{2\pi \sqrt{a^{2}+b^{2}}}\]

2. \[\frac{\mu_{0}iab}{2\pi \sqrt{a^{2}+b^{2}}}\]

3. \[\frac{\mu_{0}i\left( a+b \right)}{\pi \sqrt{a^{2}+b^{2}}}\]

4. \[\frac{2\mu_{0}i\sqrt{a^{2}+b^{2}}}{\pi ab}\]

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1 2 3 Next »

Magnetic Effects of Current

1. Current Distribution in the Outer Shell

2. Net Enclosed Current at Radius rr

3. Magnetic Field at Radius rr

Final Answer:

2. Net Enclosed Current at Radius $r$

3. Magnetic Field at Radius $r$