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 ampere is that constant current which, when maintained in each of two infinitely long, straight, parallel conductors of negligible circular cross-section, placed 1 metre apart in a vacuum, produces a force of 2 × 10⁻⁷ N per metre of length between them.

The rotational effect experienced by a current-carrying loop placed in a uniform magnetic field is called torque.

A vector quantity that measures the strength and orientation of a current loop as a magnetic source is called the magnetic dipole moment.

Deflection produced per unit current.

Deflection produced per unit voltage.

The current required to produce a unit deflection (1 division) on the scale.

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.

A Moving Coil Galvanometer (MCG) is a sensitive electromagnetic instrument used to detect and measure small electric currents (of the order of microamperes to milliamperes) by measuring the deflection of a current-carrying coil placed in a uniform magnetic field.

Combining the four dependencies: dB ∝ \[\frac {I dl sin θ}{r^2}\]​

Introducing the constant of proportionality \[\frac {μ_0}{4π}\]​​:

\[dB=\frac{\mu_0}{4\pi}\cdot\frac{Idl\sin\theta}{r^2}\]

\[{d\vec{B}=\frac{\mu_0}{4\pi}\cdot\frac{Id\vec{l}\times\hat{r}}{r^2}=\frac{\mu_0}{4\pi}\cdot\frac{Id\vec{l}\times\vec{r}}{r^3}}\]

For a finite conductor, integrate over the entire length:

\[{\vec{B}=\frac{\mu_0I}{4\pi}\int\frac{d\vec{l}\times\hat{r}}{r^2}}\]

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

\[m=NIA\]

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

Also written as:

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

VS = \[\frac{\phi}{V}=\frac{NAB}{CG}\]

where G = resistance of the galvanometer coil.

Unit: div/V

k = \[\frac{I}{\phi}=\frac{C}{NAB}\]

k is the reciprocal of current sensitivity. A galvanometer with a smaller figure of merit is more sensitive.

CS = \[\frac{\phi}{I}=\frac{NAB}{C}\]

Unit: div/A or div/μA

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.

Statement

The line integral \[\oint\vec{B}\cdot d\vec{l}\] taken around any closed loop equals μ₀ times the net steady current passing through the loop.

Proof (for a long straight wire)

• Consider an infinitely long straight wire carrying current I.

• By Biot–Savart law, field at distance r:
  B = \[\frac{\mu_0I}{2\pi r}\]

• Choose a circular Amperian loop of radius r, concentric with the wire.

• By symmetry, B is constant in magnitude and tangential (parallel to \[d\vec l\]) everywhere:
  \[\oint\vec{B}\cdot d\vec{l}=B\oint dl\] = B(2πr)

• Substituting B:
  \[\oint\vec{B}\cdot d\vec{l}=\frac{\mu_0I}{2\pi r}(2\pi r)\] = μ₀​I

Conclusion

\[\oint\vec{B}\cdot d\vec{l}=\mu_0I\]
The result is independent of the loop's radius, confirming the law's validity.

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.

Step 1: Torque due to current (Deflecting Couple):

• When current I flows through a coil of N turns, area A, in a field B: τ_deflecting = N I A B (Since radial field: sin⁡90° = 1)

Step 2: Restoring Torque (Spring):

• The phosphor-bronze strip/spring opposes the deflection. If ϕ is the angular deflection and C (or k) is the torsional constant of the spring, τ_restoring = Cϕ

Step 3: Equilibrium Condition:

• At equilibrium, deflecting torque = restoring torque: NIAB = Cϕ

Step 4: Current–Deflection Relationship:

• 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.

• 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.
• \[I_g=\frac{V}{G+R}\]