Magnetism Concept Page - 3

Definition

Calculate magnetic induction at a point along the equator of a bar magnet

NS is the bar magnet of length 2l$2l$ and pole strength m$m$. P$P$ is a point on the equatorial line at a distance d from its mid point O$O$.
Magnetic induction (B1)$(B_1)$ at P$P$ due to north pole of the magnet,
B1=1/4.(mNP2$B_1=1/4.({\displaystyle \frac{m}{N{P}^2}}$ along NP
        = 1/4.(m(d2+l2)$1/4.({\displaystyle \frac{m}{(d^2+l^2)}}$ along NPResolving B1$B_1$ and B2$B_2$ into their horizontal components.
B=B1cosθ+B2cosθ$B=B_{1}cos\theta +B_{2}cos\theta$

Definition

Lorentz force

The total force on a charge q$q$ moving with velocity v$v$ in the presence of magnetic and electric fields B$B$ and E$E$, respectively is called the Lorentz force. It is given by the expression:
F=q(v×B+E)$F=q(v\times B+E)$
The magnetic force q(v×B)$q(v\times B)$ is normal to v$v$ and work done by it is zero.

Definition

Fleming's left hand rule

Fleming's left hand rule is used to determine the direction of force exerted on a current carrying wire placed in a magnetic field. If the thumb, index finger (along magnetic field) and middle finger(along current) are held mutually perpendicular as shown in the figure, then thumb gives the direction of force on the wire.

Example

Net force on a moving electric charge

F=q[E+v×B]$F=q[E+v\times B]$

An electron passes undeflected through perpendicular electric and magnetic fields of intensity 3.4×103V/m$3.4\times {10}^{3}V/m$ and 2×10−3Wb/m2$2\times {10}^{-3}Wb/m^2$ respectively. Then its  velocity is given by:For undeflected path
F=q(V×B+E)=0$F=q(V\times B+E)=0$
or VB=|E|$VB=|E|$

V=$V=$ |E|B${\displaystyle \frac{|E|}{B}}$
   = 3.4×1032×10−3${\displaystyle \frac{3.4\times {10}^3}{2\times {10}^{-3}}}$
   = 1.7×106ms$1.7\times {10}^6{\displaystyle \frac{m}{s}}$.

Example

Energy of a charged particle moving in a magnetic field

Calculate the energy in keV of an electron (mass 9×10−31kg$9\times {10}^{-31}kg$ and charge 1.6×10−19C$1.6\times {10}^{-19}C$) moving at a speed of 3×107m/s$3\times {10}^{7}m/s$ in a magnetic field of 6×10−4T$6\times {10}^{-4}T$ perpendicular to it
E=12mv2=129×10−31kg×9×1014m2/s2=2.5 keV$E={\displaystyle \frac{1}{2}}m{v}^2={\displaystyle \frac{1}{2}}9\times {10}^{-31}kg\times 9\times {10}^{14}{m}^2/s^2=2.5keV$

Definition

Motion of a particle in a magnetic field

The particle will describe a circle if v$v$ and B$B$ are perpendicular to each other .If velocity has a component along B$B$, this component remains unchanged as the motion along the magnetic field will not be affected by the magnetic field. The motion in a plane perpendicular to B$B$ is as before a circular one, thereby producing a helical motion

Definition

Aurora Boreails

Interaction of the terrestrial magnetic field with particles from the solar wind sets up the conditions for the aurora phenomena near the poles.

The Aurora is an incredible light show caused by collisions between electrically charged particles released from the sun that enter the earths atmosphere and collide with gases such as oxygen and nitrogen. The lights are seen around the magnetic poles of the northern and southern hemispheres.

Auroras that occur in the northern hemisphere are called Aurora Borealis or northern lights and auroras that occur in the southern hempishere are called Aurora Australis or southern lights.

Definition

Motion of charged particle in combined electric and magnetic fields

Depending on the initial velocity of the charged particle the trajectory of motion in the crossed electric and magnetic fields can be trochoid, cycloid or even a straight line.

Formula

Pitch of a hellical path of a particle moving in a magnetic field

If there is a component of the velocity parallel to the magnetic field (denoted by v||$v_{||}$), it will make the particle move along the field and the path of the particle would be a helical one. The distance moved along the magnetic field in one rotation is called pitch p$p$.
p=v||T=2πmv||qB$p=v_{||}T={\displaystyle \frac{2\pi m{v}_{||}}{qB}}$

Formula

Radius of curvature of a particle moving in a magnetic field

r=mvqB$r={\displaystyle \frac{mv}{qB}}$

Definition

Example of motion of electron in external electric field

Suppose the electric filde be as shown in the figure from positive to negative plate. And an electron moves inperpendicular to the electric field as shown. The acceleration of the electron will be opposite to the electric field. Hence it will deflect downwards as shown.

Example

Motion of electron in external electric and magnetic field

Suppose an electron is left from rest from a place where both electric and magnetic field both exist. The magnetic field and electric field are perpendicular to each other. The elctron will accelerate opposite to the direction of electric field. As well the electron will also move in circular motion. Hence the motion of the electron wil be as shown in the figure. This path is known as helical path.

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