Revision: Magnetic Effects of Current and Magnetism >> Moving Charges and Magnetism Physics (Theory) ISC (Science) ISC Class 12 CISCE

The phenomenon in which a current-carrying conductor produces a magnetic field around itself is called the magnetic effect of electric current.

A current-carrying conductor is always associated with a magnetic field around it is called the magnetic effect of current. It was first discovered by Hans Christian Oersted in 1820.

Oersted's Experiment demonstrated that a current-carrying conductor produces a magnetic field around itself, proving that moving electric charges (current) generate magnetism.

A magnetic field is the region of space around a magnet or a current-carrying conductor within which a magnetic force can be experienced by another magnet or moving charge.

SI Unit: Tesla (T)
Nature: Vector quantity

The process of stopping the magnetic field from entering a region is called magnetic shielding.

The number of magnetic field lines crossing unit area kept normal to the direction of field lines is called magnetic flux density. Its unit is Wb/m²

A region of space is said to have a magnetic field if a moving charged particle experiences a deflecting force in that region. This field is denoted by \[\vec{B}\] and is also called magnetic induction.

The total force experienced by a charged particle moving with velocity \[\vec v\] in the presence of both an electric field \[\vec E\] and a magnetic field \[\vec B\] simultaneously.

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 long, cylindrical coil consisting of a large number of closely wound circular turns of insulated copper wire, in which a magnetic field is produced when current flows through it.

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."

A constant representing the ability of vacuum to support magnetic field lines; its value is 4π × 10⁻⁷N A⁻².

The magnetic field at a perpendicular distance r from a long straight conductor carrying current I₁​ is given by:

\[B_1=\frac{\mu_0I_1}{2\pi r}\]

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

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.

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

Deflection produced per unit voltage.

Deflection produced 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.

A galvanometer is said to be sensitive if it produces a large deflection even for a small current passing through it.

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.

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

dE = \[\frac{1}{4\pi\epsilon_0}\frac{dq}{r^2}\]

r is the distance from the charge element

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

θ is the angle between the current element and the line joining it to point P

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

Unit: div/A or div/μA

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.

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

where G = resistance of the galvanometer coil.

Unit: div/V

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.

The line integral of the magnetic field over a closed-loop is μ₀ times the net current enclosed by loop.

`oint_C vec B * vec l = mu_circ I_"enclosed"`

`I_"enclosed"` → net current in the closed-loop C.

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)`

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 magnetic field is the region around a magnet or current-carrying conductor where its effect can be detected (shown by the pin/iron-filing activity).
• Oersted (1820) showed that current produces a magnetic effect.
• Faraday introduced the term "magnetic lines of force."
• The magnetic field is a vector quantity; the compass-needle convention gives its direction.
• Significance: electric motors, generators, Earth's magnetism, and electromagnetism as a fundamental force.

• A cyclotron uses a small alternating electric field for repeated acceleration and a constant magnetic field to bring the particle back for the next push
• The time for one semicircular pass is independent of speed — this is the entire secret behind why the fixed-frequency voltage keeps working as the particle speeds up
• Resonance condition: applied frequency = qB / 2πm
• Maximum kinetic energy depends on the square of the dee radius and the square of the magnetic field: K = q²B²R² / 2m
• Cyclotrons cannot accelerate electrons or neutral particles
• Real machines correct for relativistic effects using synchro-cyclotrons (frequency decreases as mass increases)

• 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.

• 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.