Magnetism Concept Page - 8

Definition

Define Magnetic length and its relation with geometric length

Magnetic length is the shortest distance between the two poles of a magnet.

Magnetic length =0.84×$=0.84\times$ geometric length

Definition

Magnetometer

Magnetometers are measurement instruments used for two general purposes: to measure the magnetization of a magnetic material like a ferromagnet, or to measure the strength and, in some cases, the direction of the magnetic field at a point in space.There are two basic types of magnetometer measurement. Vector magnetometers measure the vector components of a magnetic field. Total field magnetometers or scalar magnetometers measure the magnitude of the vector magnetic field.
Superpostion of Magnetic Fields(B and BH$B_H$,Tangent):
When a freely suspended magnet is subjected  to an external magnetic field B perpendicular to BH${}_H$. Then the needle comes to rest making an angle θ$\theta$ with BH${}_H$.then B=BHtanθ$B=B_H\tan \theta$. This is known as Tangent law.Deflection Magnetometer:
Tangent law is the main principle used in Deflection.From the above concept, we can write BH=Bcotθ$B_H=B\cot \theta$
To get the sensitivity, we differentiate the above  equation and write
dBH=−Bcsc2θdθ$d{B}_H=-B{\csc }^2\theta d\theta$
So we get dBHBH=−csc2θdθcotθ=−2dθsin2θ${\displaystyle \frac{d{B}_H}{B_H}}={\displaystyle \frac{-{\csc }^2\theta d\theta }{\cot \theta }}={\displaystyle \frac{-2d\theta }{\sin 2\theta }}$
To minimize the error, we have to minimize the absolute value of dBHBH${\displaystyle \frac{d{B}_H}{B_H}}$
That will happen only when sin2θ=1$\sin 2\theta =1$ i.e. θ=45∘$\theta ={45}^{\circ }$

Law

Ampere's circuital law

Ampere's circuital law states that ∮B⃗ .dl→$\oint \overrightarrow{B}.\overrightarrow{dl}$ of the resultant magnetic field along a closed, plane curve is equal to μ0${\mu }_0$ times the total current crossing the area bounded by the closed curve provided the electric field inside the loop remains constant.
Thus, ∮B⃗ .dl→=μ0i$\oint \overrightarrow{B}.\overrightarrow{dl}={\mu }_{0}i$

Definition

Amperian loop

Ampere's circuit law uses Amperian loop to find magnetic field in a region. The Amperian loop is one such that at each point of the loop, either:

1. B$B$ is tangential to the loop and is a non zero constant, or
2. B$B$ is normal to the loop, or 
3. B$B$ vanishes

where B$B$ is the induced magnetic field.

Definition

Force on a wire in magnetic field

When the magnetic field (B$B$) and the current carrying wire of length (L$L$) are perpendicular, then force on the wire is given by:
F=BIL$F=BIL$

Example

Force acting on various geometrical configurations of current carrying conductors in uniform magnetic field

A conductor in the form of a right angle ABC with AB=3 cm$AB=3cm$ and BC=4 cm$BC=4cm$ carries a current of 10$10$  A. There is a uniform magnetic field of 5 T$5T$ perpendicular to the plane of the conductor. The force on the conductor will be:F=Bil$F=Bil$
=5×10×5×10−2N$5\times 10\times 5\times {10}^{-2}N$
=2.5 N$2.5N$

Example

Torque acting on a loop in a uniform magnetic field

A rectangular loop carrying a current i$i$ is placed in a uniform magnetic field B$B$. The area enclosed by the loop is A$A$. If there are n$n$ turns in the loop, the torque acting on the loop is given by:
τ⃗ =M⃗ ×B⃗ $\overrightarrow{\tau }=\overrightarrow{M}\times \overrightarrow{B}$
=ni(A⃗ ×B⃗ )$=ni(\overrightarrow{A}\times \overrightarrow{B})$    (∵M⃗ =niA⃗ )$(\because \overrightarrow{M}=ni\overrightarrow{A})$

Definition

Force on current carrying conductor in magnetic field

An electric current flowing through a conductor produces a magnetic field. The field so produced exerts a force on a magnet placed in the vicinity of a conductor. The magnet also exerts an equal and opposite force on the current-carrying conductor.
The displacement of the rod in the diagram suggests that a force is exerted on the current-carrying aluminium rod when it is placed on a magnetic field.Here the direction of force is also reversed when the direction of current through the conductor is reversed. The direction of force on the conductor depends upon the direction of current and the direction of magnetic field.The displacement of the rod is maximum when the direction of current is at right angles to the direction of the magnetic field.   

Definition

Analogy between a bar magnet and a solenoid

A bar magnet may be thought of as a large number of circulating currents in analogy with a solenoid. On cutting a bar magnet in half we get two smaller solenoids with weaker magnetic properties.The field lines remain continuous, emerging from one face of the solenoid and entering into the other face. (i.e. similar magnetic fields). The magnetic moment of a bar magnet is also equal to the magnetic moment of an equivalent solenoid. 

Definition

Equivalence between magnetic dipole and a solenoid

 For current I$I$  through a solenoid with total number of turns N$N$ and with cross-sectional area A$A$, the magnetic dipole moment of the solenoid is given by
ms=NIA$m_s=NIA$...................... (i)
The magnetic dipole moment of a bar magnet of pole strength, m$m$, and length, 2l$2l$, is given by
mb=2ml$m_b=2ml$.......................(ii)
By analogy between solenoid and bar magnet, the pole strength, ms$m_s$, of the solenoid can be obtained using the above two equations as:
2msl=NIA$2m_{s}l=NIA$
ms=NIA/2l=nIA$m_s=NIA/2l=nIA$ , where 
n=N/2l$n=N/2l$=no. of turns per unit length of solenoid.
Thus Pole-strength of solenoid =(Number of turns per unit length)*(electric current)*(cross-sectional area of solenoid). 

Definition

Magnetic field lines of a bar magnet, current carrying finite solenoid and an electric dipole

The magnetic field lines of a magnet (or a solenoid) form continuous closed loops,emerging from one face of the solenoid and entering into the other face.This is unlike the electric dipole where these field lines begin from a positive charge and end on the negative charge or escape to infinity.

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