Magnetic effect of current

 

Discovered by Oerested.

A stationary charge produces only electric filed. But moving charge produces both electric and magnetic field.

When electric current is passed through a conductor then only magnetic field is observed as conductor is electrically neutral so electrical field is not observed.

Strength of magnetic field is called magnetic induction or magnetic flux density.

Magnetic field induction is vector.

Magnetic field lines always form closed loop.

Two field lines do not intersect.

A tangent drawn at a point on field line gives the direction of the magnetic field at that point.

SI unit of magnetic field induction B is tesla (T).

Biot Savart Law

According to this law, the magnetic field induction at a point due to small piece of current carrying conductor is given by

$dB=\dfrac{µ Idlsin\theta}{4\pi r^2}$

Where B is the magnetic field at any point at a distance r from the wire, I is the current through the wire and $µ= µ_0 µ_r$, where $µ_0 =4π10e-7$ Henry/meter.

$\to$ Magnetic field on the axis of the conductor is $0$.

$\textbf{Magnetic field due to flow of current}$: $\textbf{Through straight conductor}$:

$B=\dfrac{µ I(sin\alpha + sin\beta)}{4\pi r}$

Where $\alpha$ and $\beta$ are angles made by the lower and upper end of the linear conductor with perpendicular line.

$\to$ When the conductor is of infinite length then $B=\dfrac{µI}{2\pi r}$

For semi infinite wire $B=\dfrac{µI}{4\pi r}$

When point lies at conductor $B=0$

$\to$ Direction of magnetic lines of force can be given by Right Hand Thumb Rule.

$\textbf{Magnetic field at the Center of the Circular Coil Carrying conductor}$:

$B=\dfrac{µNI}{2r}$ where $N$ is the number of turns.

$\textbf{Magnetic field at a point on axis of circular current carrying coil}$ is :

$B=\dfrac{µNIr^2}{2(x^2+r^2 )^{3/2}}$

Where r is the radius of the coil and x is the distance of the point from the center of coil.

$\to$ If $r>>x$ then $B=\dfrac{µNI}{2r}$

$\to$ If $x>>>r$ then $B=\dfrac{µINr^2}{x^3}$

$\textbf{Magnetic field Due to solenoid carrying current}$:

$B=\dfrac{µnI(cos\alpha - cos\beta)}{2}$, where $\alpha$ and $\beta$ are angle subtended by ends of solenoid at a point on the axis.

At center it is $µnI$ and at end it is $µ\dfrac{nI}{2}$

Magnetic field due to moving charge in circular orbit at the center:

$B=\frac{\mu qV}{4\pi r^2}$

Magnetic field induction at centre $O$ of the circular arc of Radius $R$ lying in the plane of paper is

$B=\dfrac{µIα}{4\pi r}$ , Direction of $B$ is normal to the plane of paper downward.

The $\textbf{magnetic field induction of two common centered}$ coils:

When current is flowing in same direction is

$B_{net}=B_1+B_2=\dfrac{µ}{2}(\dfrac{N_1}{r_1} + \dfrac{N_2}{r_2})$

If current was flowing in opposite direction:

$B_{net}=B_1-B_2=\dfrac{µ}{2}(\dfrac{N_1}{r_1} - \dfrac{N_2}{r_2})$

Lorentz Force on a Charged Particle in Uniform magnetic field

When a charge $q$ is projected in the uniform magnetic field $B$ with velcotity $v$ then it experiences a magnetic force $F$ which is given by $F=q(v*B)=Bqvsin\alpha$

Where $\alpha$ is the angle between $v$ and $B$.

The direction of Lorentz force can be calculated using Fleming’s Left hand rule or Right hand palm rule.

If a charged particle is projected along or opposite to the magnetic field then the angle between them is either $0$ or $180$ which results in $sin\alpha =0$, and $F=0$. Moreover, Speed, Velocity, K.E and momentum remain constant. And it doesn’t experience any force.

If a charged particle is projected along or opposite to a magnetic field then charge will travel in a straight path with no deflection with constant speed kinetic energy, velocity and momentum.

Charge will travel in a helical path if $\alpha$ is other than $0$,$90$,$180$ degree.

Lorentz Force on a charged particle moving in electric and magnetic field.

If a charged particle of charge $q$ is moving with velocity $v$ is subjected to electric field $E$ and magnetic field $B$ then net Lorentz force is given by .

$F_{net}=q(E+v*B)$

Force on current carrying wire kept in magnetic field.

If a current carrying straight wire of length $l$ is kept in magnetic field $B$ then the force experienced is given by

$F=I(L*B)=BILsin\alpha$ , If $\alpha$ is $0$ or $180$ degrees, $F=0$.

Magnetic Dipole Moment:

Product of current and area of the loop is called magnetic dipole moment.

$M=N.I.A$

If point charge revolve around the circular path then, $M=IA=\dfrac{qvr}{2}=\dfrac{qwr^2}{2}$