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.

A toroid is a hollow circular ring (like an anchor ring) on which a large number of turns of insulated wire are closely wound. It is essentially a straight solenoid bent into a closed circular shape, forming an "endless solenoid."

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.

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

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

Deflection produced per unit current.

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.

Deflection produced per unit voltage.

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

A device used to measure the potential difference (voltage) between two points in a circuit. It is connected in parallel across those points and ideally offers infinite resistance so that it does not alter the current in the circuit.

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

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

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

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:

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

\[m=NIA\]

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

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)`

Stretch the index finger, the middle finger, and the thumb of the left hand mutually perpendicular to each other. If the index finger is in the direction of the magnetic field and the middle finger points in the direction of the current, then the thumb will point towards the direction of the force 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:

• A circular current loop produces a magnetic field whose axial value is B = \[\frac{\mu_0IR^2}{2(x^2+R^2)^{3/2}}\].
• At the centre of the loop (x = 0), this simplifies to B₀ = \[\frac {μ_0I}{2R}\]​, and for N turns, B₀ = \[\frac {μ_0NI}{2R}\].
• Perpendicular field components from opposite points on the loop cancel; only axial components add up.
• Direction follows the right-hand thumb rule; one face of the loop acts as a north pole, the other as a south pole.
• Straight wire segments (as in a semicircular arc problem) contribute zero field at a point lying on the line of the wire itself.

• Biot–Savart's Law gives the field due to a current element; integrating over a finite wire gives B = \[\frac {μ_0I}{4πr}\](sin ϕ₁ + sin ϕ₂).
• For an infinite wire, B = \[\frac {μ_0I}{2πr}\]​.
• For a semi-infinite wire, B = \[\frac {μ_0I}{4πr}\] (half of infinite wire).
• For a point on the perpendicular bisector of a finite wire of length l, B = \[\frac{\mu_0I}{2\pi r}\cdot\frac{l}{\sqrt{4r^2+l^2}}\].
• The field is always directly proportional to I and inversely proportional to r.
• Direction is found using the right-hand palm rule.

• A current-carrying conductor in a magnetic field experiences a force perpendicular to both the current and the field direction.
• Reversing current or reversing field polarity reverses the force direction.
• Formula: F = BIl sin ⁡θ; vector form \[\vec F\] = I\[\vec l\] × \[\vec B\].
• Force is zero when the wire is parallel to \[\vec{B}\] and maximum (F_max = BIl) when perpendicular.
• Fleming's Left-Hand Rule (thumb = force, forefinger = field, middle finger = current) is the standard tool for direction in Indian board exams.