Revision: Class 12 >> Magnetic Effect of Current NEET (UG) Magnetic Effect of Current

Current passed through each of the two infinitely long parallel straight conductors kept at a distance of one meter apart in vacuum causes each conductor to experience a force of 2 × 10^-7 newton per meter length of the conductor.

An insulated long wire closely wound in the form of a helix, whose length is very large compared to its diameter, is called a solenoid.

An anchor ring (torus) around which a large number of turns of metallic wire are wound, forming an endless solenoid, is called a toroid.

The current sensitivity of a galvanometer is defined as the deflection produced in the galvanometer when a unit current flows through it.  
Mathematically, it can be given by:

I_S = `(NBA)/k`

Where k is the couple per unit twist.

Current sensitivity is defined as the deflection e per unit current.

F = IL × B

\[\vec{B}=\frac{\mu_0IR^2}{2(x^2+R^2)^{3/2}}\hat{i}\]

Where:

• I = current
• R = radius of loop
• x = distance from centre along axis
• μ0 = permeability of free space

\[B=\frac{\mu_0I}{2\pi r}\]

\[B=\frac{\mu_0Ir}{2\pi R^2}\]

\[B=\frac{\mu_0I}{2\pi r}\]

\[B=\mu_0nI\]

\[n=\frac{N}{L}\] (number of turns per unit length)

\[F=qvB\sin\theta\]

Vector form:

\[\vec{F}=q(\vec{v}\times\vec{B})\]

Special cases:

\[v\parallel B\Rightarrow F=0\]

\[v\perp B\Rightarrow F=qvB\]

Centripetal force provided by magnetic force: \[\frac{mv^2}{r}=qvB\]

Angular speed = \[\omega=\frac{qB}{m}\]

Period of circular motion = \[T=\frac{2\pi m}{qB}\]

Frequency = \[f=\frac{qB}{2\pi m}\]

\[F=BIL\sin\theta\]

Vector Form:

\[\vec{F}=I(\vec{L}\times\vec{B})\]

Special Cases:

• θ = 90^∘ → F = BILF 
• θ = 0^∘ → F = 0

\[F=\frac{\mu_0I_1I_2}{2\pi d}\times l\]

Per unit length:

\[\frac{F}{l}=\frac{\mu_0I_1I_2}{2\pi d}\]

Force acts along the line joining the wires

\[\tau=NIAB\sin\theta\]

Also written as:

\[\vec{\tau}=\vec{m}\times\vec{B}\]

\[m=NIA\]

\[\mathrm{V.S.}=\frac{\theta}{V}=\frac{NBA}{CR}\]

\[\mathrm{C.S.}=\frac{\theta}{I}=\frac{NBA}{C}\]

The magnitude of magnetic induction (dB) at a point due to a small element of current carrying conductor is:
(i) directly proportional to current (dB ∝ I),
(ii) directly proportional to length of element (dB ∝ dl),
(iii) directly proportional to sine of angle between element and line joining its centre to the point (dB ∝ sin θ),
(iv) inversely proportional to square of distance (dB ∝ 1/r²).

Applications

• Magnetic field at centre of circular coil.
• Magnetic field on axis of the coil.
• Magnetic field at a distance from a straight current-carrying conductor.

Maxwell’s cork screw rule

If a straight current-carrying conductor is held in the right hand such that the thumb points in the direction of current, then the curled fingers give the direction of the magnetic field lines around the conductor.

Right-hand thumb rule

If a right-handed screw is rotated in such a way that it moves forward in the direction of current through a conductor, then the direction of rotation of the screw gives the direction of the magnetic field lines around the conductor.

This law states that the line integral of magnetic field density (B) along an imaginary closed path is equal to the product of the current enclosed by the path and the permeability of the medium.

\[\oint\vec{B}.\overrightarrow{dl}=\mu_{0}I\]

The line integral of magnetic field of induction \[\vec B\] around any closed path in free space equals μ₀​ times the total current through the area bounded by the path.
∮\[\vec B\] ⋅ \[\vec d\]s = μ₀​I. The closed loop is called an Amperian loop; I is the net current enclosed.

Applications

• Magnetic field due to a long straight current-carrying wire.
• Magnetic field inside an ideal long straight solenoid.
• Magnetic induction along the axis of a toroid.

The toroid is a solenoid bent into the shape of a hollow doughnut.

According to Ampere's circuital law.

`phivecB.vec(dL) = mu_0I`

Here current 'I' flow through the ring as many times as there are the N no. of turns.

∴ `phivecB.vec(dL) = mu_0NI` ......(1)

Now, B and dL are in the same direction.

∴ `phivecB.vec(dL) = BphidL`

∴ `phivecB.vec(dL) = B.(2pir)` .....(2)

From (1) and (2),

`mu_0NI = B.(2pir)`

∴ B = `(mu_0NI)/(2pir)`

If the thumb, forefinger and middle finger of the left hand are stretched mutually perpendicular to each other, and the forefinger points in the direction of the magnetic field and the middle finger points in the direction of the current, then the thumb gives the direction of the force acting on the conductor.

• B inside is independent of length and diameter and is uniform across the cross-section.
• B at the ends = ½ × B at the centre.
• B outside the solenoid is zero.
• Field pattern is similar to that of a bar magnet.
• B inside the toroid is independent of r, provided turns per unit length remain same.
• B outside the toroid is zero; field is confined to the core on which winding is made.

• A current-carrying conductor placed in a magnetic field experiences a force when the direction of current is not parallel to the magnetic field.
• The direction of force reverses when the direction of current or the direction of magnetic field is reversed, and no force acts when current flows parallel to the magnetic field.

Based on the torque on the current-carrying coil in the magnetic field: \[\tau=NIAB\]

Restoring torque: \[\tau=C\theta\]

At equilibrium: \[NIAB=C\theta\]

• A galvanometer is converted into an ammeter by connecting a low resistance (shunt) in parallel with it.
• Only a small current flows through the galvanometer, and the remaining current flows through the shunt.
• Total current: \[I=I_g+I_s\]
• Ideal ammeter has very low (≈ 0) resistance.

• A galvanometer is converted into a voltmeter by connecting a high resistance in series with it.
• The scale is calibrated in volts.
• \[I_g=\frac{V}{G+R}\]