Name: Biot Savart and Ampere's Law: A Review
Uploaded: 2019-07-19
Duration: 21 min 6 s
Description: In this video, let us review the Biot-Savart law, Ampere's Law, the right-hand rule for a magnetic field, and how to pick an amperian loop in a magnetic field.

Question

Write Ampere's circuital law. Find the expression for magnetic field in long current carrying solenoid. Draw required diagram.

Accepted Answer

Ampere's circuital law states that line integral of magnetic field produced by electric currents in air or vacuum over a closed path enclosing an area, is equal to $μ_{0}$ times the algebraic sum of currents passing through that area.

$\oint \vec{B} . d \vec{l} = μ_{0} \sum_{} I$

Magnetic field inside an infinitely long current carrying solenoid.

Consider a current carrying long solenoid. Let $n$ be the turns density, i.e, number of turns per unit length of the solenoid and $I$ be the current flowing through the solenoid and the turns of the solenoid be closely packed.

Consider a rectangular amperian loop $P Q R S$ near the middle of the solenoid, where $P Q = x$ . Let the magnetic field along the path $P Q$ be $B$ and is zero along $R S$ . As the paths $Q R$ and $S P$ are perpendicular to the axis of the solenoid, the magnetic field component along these paths is zero. The total number of turns in the length $x$ is $n x$ .

From Ampere's circuital law

$\oint_{PQRS} \vec{B} \cdot d \vec{l} = μ_{0} \sum_{} I$

$\oint_{PQRS} \vec{B} \cdot d \vec{l} = \int_{P}^{Q} \vec{B} \cdot d \vec{l} + \int_{Q}^{R} \vec{B} \cdot d \vec{l} + \int_{R}^{S} \vec{B} \cdot d \vec{l} + \int_{S}^{P} \vec{B} \cdot d \vec{l} = μ_{0} \sum_{} I$

$\int_{P}^{Q} \vec{B} \cdot d \vec{l} = \int_{P}^{Q} B d l \cos 0 ° = B x$ $(∵ \int_{P}^{Q} d l = x)$

$\int_{Q}^{R} \vec{B} \cdot d \vec{l} = \int_{Q}^{R} B d l \cos 90 ° = 0; \int_{S}^{P} \vec{B} \cdot d \vec{l} = \int_{S}^{P} B d l \cos 90 ° = 0$

$\int_{R}^{S} \vec{B} \cdot d \vec{l} = 0$ ( $∵ B = 0$ outside the solenoid)

$\oint_{PQRS} \vec{B} \cdot d \vec{l} = B x + 0 + 0 + 0 = μ_{0} \sum_{} I$

$B x = μ_{0} \sum_{} I$

But $\sum_{} I = N I = n x I$

$B x = μ_{0} n x I$

$B = μ_{0} n I$

Where $n$ is the turns density.

If a material of permeability $μ$ is placed inside the solenoid, then

$B_{m} = μ n I = μ_{r} μ_{0} n I$

$μ_{r}$ is the relative permeability of the material.

The magnetic field in an infinitely long solenoid is uniform throughout the space inside the solenoid. If the solenoid is finite but the diameter of the solenoid is much less than its length, then the result is valid inside and far away from the ends.

Write Ampere's circuital law. Find the expression for magnetic field in long current carrying solenoid. Draw required diagram.

Important Questions on Moving Charges and Magnetism

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