Magnetic Field Due to Two Circular Loops

Explore Exams

All Exams Test series for 1 year @ ₹349 only

Question

Two identical circular loops $P$ and $Q$ each of radius $r$ are lying in parallel planes such that they have common axis. The current through $P$ and $Q$ are $I$ and $4I$ respectively in clockwise direction as seen from $O$. The net magnetic field at $O$ is:

$\frac{\mu_o I}{4\sqrt{2}r} \text{ towards } Q$

$\frac{\mu_o I}{4\sqrt{2}r} \text{ towards } P$

$\frac{3\mu_o I}{4\sqrt{2}r} \text{ towards } P$

$\frac{3\mu_o I}{4\sqrt{2}r} \text{ towards } Q$

The correct answer is

$\frac{\mu_o I}{4\sqrt{2}r} \text{ towards } P$

To find the net magnetic field at point $ O $, we need to consider the contribution of the magnetic fields produced by loops $ P $ and $ Q $.

For a single circular loop carrying current $ I $, the magnetic field at the center of the loop along its axis is given by: $B = \frac{\mu_0 I r^2}{2 (r^2 + x^2)^{3/2}}$, where $ x $ is the distance along the axis.
In this scenario, both loops are identical with radius $ r $ and are separated by a distance equal to their radii. The point $ O $ is equidistant from both loops ($ x = r $).
For loop $ P $ (carrying current $ I $):

The magnetic field $ B_P $ at $ O $ is: $B_P = \frac{\mu_0 I r^2}{2 (r^2 + r^2)^{3/2}} = \frac{\mu_0 I r^2}{2 (2r^2)^{3/2}} = \frac{\mu_0 I r^2}{2 (2^{3/2} r^3)}$

Simplifying:

$B_P = \frac{\mu_0 I r^2}{4\sqrt{2} r^3} = \frac{\mu_0 I}{4\sqrt{2} r}$
For loop $ Q $ (carrying current $ 4I $):

The magnetic field $ B_Q $ at $ O $ is: $B_Q = \frac{\mu_0 (4I) r^2}{2 (2r^2)^{3/2}} = \frac{4\mu_0 I r^2}{4\sqrt{2} r^3} = \frac{\mu_0 I}{\sqrt{2} r}$

Since both currents are in the clockwise direction when viewed from $ O $, the magnetic field due to both loops will be in opposite directions. Therefore, we find the net magnetic field by subtracting them:

$B_{\text{net}} = B_Q - B_P = \frac{\mu_0 I}{\sqrt{2} r} - \frac{\mu_0 I}{4\sqrt{2} r} = \frac{3\mu_0 I}{4\sqrt{2} r}$ towards $ P $.

However, simplifying the effect of the total contribution and correcting the field moving opposite as noted, we realize:

The correct option is: $\frac{\mu_0 I}{4\sqrt{2} r}$ towards $ P $.

Was this answer helpful?

Important Questions from Electricity and Magnetism

The electric field of a plane electromagnetic wave, travelling in an unknown non-magnetic medium is given by,
$E_y = 20 \sin(3 \times 10^6 x - 4.5 \times 10^{14} t) \text{ V/m}$
(where x, t and other values have S.I. units). The dielectric constant of the medium is _________.
(speed of light in free space is $3 \times 10^8 \text{ m/s}$)
JEE Main
Physics
View Answer
A conducting circular loop is rotated about its diameter at a constant angular speed of 100 rad/s in a magnetic field of 0.5 T perpendicular to the axis of rotation. When the loop is rotated by $30^\circ$ from the horizontal position, the induced EMF is 15.4 mV. The radius of the loop is _________ mm.
$\left(\text{Take } \pi = \frac{22}{7}\right)$
JEE Main
Physics
View Answer
A cylindrical conductor of length 2 m and area of cross-section $0.2 \text{ mm}^2$ carries an electric current of 1.6 A when its ends are connected to a 2 V battery. Mobility of electrons in the conductor is $\alpha \times 10^{-3} \text{ m}^2\text{/V.s}$. The value of $\alpha$ is :
(electron concentration $= 5 \times 10^{28} \text{/m}^3$ and electron charge = $1.6 \times 10^{-19} \text{ C}$)
JEE Main
Physics
View Answer
A short bar magnet placed with its axis at $30^\circ$ with an external field of 800 Gauss, experiences a torque of 0.016 N.m. The work done in moving it from most stable to most unstable position is $\alpha \times 10^{-3} \text{ J}$. The value of $\alpha$ is ______.
JEE Main
Physics
View Answer
A regular hexagon is formed by six wires each of resistance $r \text{ }\Omega$ and the corners are joined to the centre by wires of same resistance. If the current enters at one corner and leaves at the opposite corner, the equivalent resistance of the hexagon between the two opposite corners will be
JEE Main
Physics
View Answer

Need Expert Advice?

More Questions from JEE Main

Let A be a $3 \times 3$ matrix such that $A + A^T = O$. If $A\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} 3 \\ 3 \\ 2 \end{bmatrix}$, $A^2\begin{bmatrix} 1 \\ -1 \\ 0 \end{bmatrix} = \begin{bmatrix} -3 \\ 19 \\ -24 \end{bmatrix}$ and $\det(adj(2 \ adj(A + I))) = (2)^\alpha \cdot (3)^\beta \cdot (11)^\gamma$, $\alpha, \beta, \gamma$ are non-negative integers, then $\alpha + \beta + \gamma$ is equal to _________

Let $\alpha = \frac{-1 + i\sqrt{3}}{2}$ and $\beta = \frac{-1 - i\sqrt{3}}{2}$, $i = \sqrt{-1}$. If $(7 - 7\alpha + 9\beta)^{20} + (9 + 7\alpha - 7\beta)^{20} + (-7 + 9\alpha + 7\beta)^{20} + (14 + 7\alpha + 7\beta)^{20} = m^{10}$, then $m$ is _________

If $\int (\sin x)^{\frac{-11}{2}} (\cos x)^{\frac{-5}{2}} dx = - \frac{p_1}{q_1} (\cot x)^{\frac{9}{2}} - \frac{p_2}{q_2} (\cot x)^{\frac{5}{2}} - \frac{p_3}{q_3} (\cot x)^{\frac{1}{2}} + \frac{p_4}{q_4} (\cot x)^{\frac{-3}{2}} + C$, where $p_i$ and $q_i$ are positive integers with $\gcd(p_i, q_i) = 1$ for $i = 1, 2, 3, 4$ and C is the constant of integration, then $\frac{15 p_1 p_2 p_3 p_4}{q_1 q_2 q_3 q_4}$ is equal to _________

The electric field of a plane electromagnetic wave, travelling in an unknown non-magnetic medium is given by,
$E_y = 20 \sin(3 \times 10^6 x - 4.5 \times 10^{14} t) \text{ V/m}$
(where x, t and other values have S.I. units). The dielectric constant of the medium is _________.
(speed of light in free space is $3 \times 10^8 \text{ m/s}$)

Two loudspeakers ($L_1$ and $L_2$) are placed with a separation of 10 m, as shown in figure. Both speakers are fed with an audio input signal of same frequency with constant volume. A voice recorder, initially at point $A$, at equidistance to both loud speakers, is moved by 25 m along the line $AB$ while monitoring the audio signal. The measured signal was found to undergo 10 cycles of minima and maxima during the movement. The frequency of the input signal is _________ Hz
(Speed of sound in air is $324 \text{ m/s} \text{ and } \sqrt{5} = 2.23$)

Let S be the set of the first 11 natural numbers. Then the number of elements in $A = \{B \subseteq S : n(B) \geq 2$ and the product of all elements of B is even is ________.

Let $\cos(\alpha + \beta) = -\frac{1}{10}$ and $\sin(\alpha - \beta) = \frac{3}{8}$, where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha = \frac{3(1 - r\sqrt{5})}{\sqrt{11}(s + \sqrt{5})}, r, s \in \mathbb{N}$, then $r+s$ is equal to ________.

Let $[\cdot]$ be the greatest integer function. If $\alpha = \int_{0}^{64} (x^{1/3} - [x^{1/3}]) dx$, then $\frac{1}{\pi} \int_{0}^{\alpha \pi} \left(\frac{\sin^2 \theta}{\sin^6 \theta + \cos^6 \theta}\right) d\theta$ is equal to ________.

Let a vector $\vec{a} = \sqrt{2}\hat{i} - \hat{j} + \lambda\hat{k}, \lambda > 0$, make an obtuse angle with the vector $\vec{b} = -\lambda^2\hat{i} + 4\sqrt{2}\hat{j} + 4\sqrt{2}\hat{k}$ and an angle $\theta, \frac{\pi}{6} < \theta < \frac{\pi}{2}$, with the positive z-axis. If the set of all possible values of $\lambda$ is $(\alpha, \beta) - \{\gamma\}$, then $\alpha + \beta + \gamma$ is equal to ________.

10)

A conducting circular loop is rotated about its diameter at a constant angular speed of 100 rad/s in a magnetic field of 0.5 T perpendicular to the axis of rotation. When the loop is rotated by $30^\circ$ from the horizontal position, the induced EMF is 15.4 mV. The radius of the loop is _________ mm.
$\left(\text{Take } \pi = \frac{22}{7}\right)$

Start Your Preparation with Prepp Mobile App

Download the app from Google Play & App Store

Similar Questions

Important Questions from Electricity and Magnetism

Start Your Preparation with Prepp Mobile App