Two identical circular wires of radius 20 cm and carrying current \(\sqrt2 \)A are placed in perpendicular planes as shown in figure. The net magnetic field at the centre of the circular wires is _________ × 10^{-8} T.
Correct Answer: 628
Approach Solution - 1
Approach Solution -2
Magnetic Field of Perpendicular Circular Loops
Step 1: Magnetic Field due to a Circular Loop
The magnetic field at the center of a circular loop of radius \( r \) carrying current \( i \) is given by:
\( B = \frac{\mu_0 i}{2r} \)
where \( \mu_0 = 4\pi \times 10^{-7} \text{ T m/A} \) is the permeability of free space.
Step 2: Magnetic Field due to Each Loop
The two loops are identical, with radius \( r = 20 \text{ cm} = 0.2 \text{ m} \) and current \( i = \sqrt{2} \text{ A} \). The magnetic field due to each loop at the center has a magnitude of:
\( B = \frac{\mu_0 i}{2r} = \frac{(4\pi \times 10^{-7} \text{ T m/A}) (\sqrt{2} \text{ A})}{2(0.2 \text{ m})} \)
Step 3: Net Magnetic Field
Since the loops are in perpendicular planes, their magnetic fields at the center are perpendicular to each other. Let the magnetic field due to one loop be along the x-axis and the magnetic field due to the other loop be along the y-axis. The net magnetic field (\( \vec{B}_{net} \)) is the vector sum of the individual fields:
\( \vec{B}_{net} = \frac{\mu_0 i}{2r} \hat{i} + \frac{\mu_0 i}{2r} \hat{j} \)
The magnitude of the net magnetic field is:
\( B_{net} = \sqrt{\left( \frac{\mu_0 i}{2r} \right)^2 + \left( \frac{\mu_0 i}{2r} \right)^2} = \sqrt{2} \frac{\mu_0 i}{2r} = \frac{\mu_0 i \sqrt{2}}{2r} \)
\( B_{net} = \frac{(4\pi \times 10^{-7} \text{ T m/A}) (\sqrt{2} \text{ A}) \sqrt{2}}{2(0.2 \text{ m})} = \frac{4\pi \times 10^{-7} \times 2}{0.4} = 2\pi \times 10^{-6} \text{ T} \)
Using \( \pi = 3.14 \):
\( B_{net} = 2 \times 3.14 \times 10^{-6} \text{ T} = 6.28 \times 10^{-6} \text{ T} = 628 \times 10^{-8} \text{ T} \)
Conclusion:
The net magnetic field at the center is \( \mathbf{628 \times 10^{-8} \text{ T}} \).
Top JEE Main Physics Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
- $I _{ CM }$ is the moment of inertia of a circular disc about an axis (CM) passing through its center and perpendicular to the plane of disc $I_{A B}$ is it's moment of inertia about an axis $AB$ perpendicular to plane and parallel to axis $CM$ at a distance $\frac{2}{3} R$ from center Where $R$ is the radius of the dise The ratio of $I _{ AB }$ and $I _{ CM }$ is $x: 9$ The value of $x$ is______
- A nucleus disintegrates into two smaller parts, which have their velocities in the ratio $3: 2$ The ratio of their nuclear sizes will be $\left(\frac{x}{3}\right)^{\frac{1}{3}}$ The value of ' $x$ ' is:-
Top JEE Main Magnetic Field Questions
- Consider a rod moving in a magnetic field as shown:
The induced emf across the ends of the rod is The magnetic flux through a loop varies with time as \(Φ= 5t^2 -3t +5\). If the resistance of loop is \(8\) , find the current through it at \(t = 2\) \(s\)
- Two rings of equal radius \(R\) arranged perpendicular to each other with common center at \(C\), carrying equal current \(I\). Find magnetic field at \(C\).
- A long wire carrying current \(\sqrt{(2)}A\) is placed in a uniform magnetic field of \(3 \times 10^{-5}\; T\). If the magnetic field is perpendicular to wire, find magnetic force on the length of wire.
- The electric field in an electromagnetic wave is moving in a free space given as vec \(E =E_ 0 \;sin(⍵t - kz)\) \(i\) The corresponding magnetic field will be:
Top JEE Main Questions
- In a hydrogen like atom, when an electron jumps from the $M$ - shell to the $L$ - shell, the wavelength of emitted radiation is $\lambda$. If an electron jumps from $N$-shell to the $L$-shell, the wavelength of emitted radiation will be :
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.