Question:
A current loop consists of two identical semicircular parts each of radius $R$, one lying in the $x-y$ plane and the other in $x-z$ plane. If the current in the loop is $i$. The resultant magnetic field due to the two semicircular parts at their common centre is
A current loop consists of two identical semicircular parts each of radius $R$, one lying in the $x-y$ plane and the other in $x-z$ plane. If the current in the loop is $i$. The resultant magnetic field due to the two semicircular parts at their common centre is
Updated On: May 2, 2024
- $\frac{\mu_0 i}{2\sqrt2 R}$
- $\frac{\mu_0 i}{2R}$
- $\frac{\mu_0 i}{4R}$
- $\frac{\mu_0 i}{\sqrt2 R}$
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The Correct Option is A
Solution and Explanation
The loop mentioned in the question must look
like one as shown in the figure.
Magnetic field at the centre due to semicircular loop lying in x-y plane, $B_{xy}=\frac{1}{2}\big(\frac{\mu_0 i}{2R}\big)$ negative z direction.
Similarly field due to loop in x-z plane,
$B_{xz}=\frac{1}{2}\big(\frac{\mu_0 i}{2R}\big)$ in negative y direction.
$\therefore$ Magnitude of resultant magnetic field,
$B=\sqrt{{B_{xy}^2}+{B_{xz}^2}}=\sqrt{\big(\frac{\mu_0 i}{4R}\big)^2+\big(\frac{\mu_0 i}{4R}\big)^2}$
$ \, \, \, \, =\frac{\mu_0 i}{4R}\sqrt2=\frac{\mu_0 i}{2 \sqrt2 R}$
like one as shown in the figure.
Magnetic field at the centre due to semicircular loop lying in x-y plane, $B_{xy}=\frac{1}{2}\big(\frac{\mu_0 i}{2R}\big)$ negative z direction.
Similarly field due to loop in x-z plane,
$B_{xz}=\frac{1}{2}\big(\frac{\mu_0 i}{2R}\big)$ in negative y direction.
$\therefore$ Magnitude of resultant magnetic field,
$B=\sqrt{{B_{xy}^2}+{B_{xz}^2}}=\sqrt{\big(\frac{\mu_0 i}{4R}\big)^2+\big(\frac{\mu_0 i}{4R}\big)^2}$
$ \, \, \, \, =\frac{\mu_0 i}{4R}\sqrt2=\frac{\mu_0 i}{2 \sqrt2 R}$
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Concepts Used:
Moving Charges and Magnetism
Moving charges generate an electric field and the rate of flow of charge is known as current. This is the basic concept in Electrostatics. Another important concept related to moving electric charges is the magnetic effect of current. Magnetism is caused by the current.
Magnetism:
- The relationship between a Moving Charge and Magnetism is that Magnetism is produced by the movement of charges.
- And Magnetism is a property that is displayed by Magnets and produced by moving charges, which results in objects being attracted or pushed away.
Magnetic Field:
Region in space around a magnet where the Magnet has its Magnetic effect is called the Magnetic field of the Magnet. Let us suppose that there is a point charge q (moving with a velocity v and, located at r at a given time t) in presence of both the electric field E (r) and the magnetic field B (r). The force on an electric charge q due to both of them can be written as,
F = q [ E (r) + v × B (r)] ≡ EElectric +Fmagnetic
This force was based on the extensive experiments of Ampere and others. It is called the Lorentz force.