Revision: Magnetic Effects of Current and Magnetism CUET (UG) Magnetic Effects of Current and Magnetism

The force experienced by a moving charge in the presence of a magnetic field, which depends on charge q, velocity v and magnetic field B, and which is opposite in direction on a negative charge compared to a positive charge, is called the magnetic force.

When both electric and magnetic fields act on a charge, the total force is called the Lorentz force.

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

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 solenoid is a coil whose length is much greater than its radius, producing a uniform magnetic field inside and nearly zero field outside.

OR

A solenoid is a long helical coil of insulated wire with many closely spaced turns, whose length l is much greater than its radius R (i.e., l ≫ R), such that it produces a strong, uniform magnetic field inside and a negligible field outside.

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 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.

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 current.

Deflection produced per unit voltage.

The temperature above which a ferromagnetic substance becomes paramagnetic is called curie temperature. 

The path in a magnetic field in which a unit north pole tends to move when allowed to do so is known as magnetic field lines of force.

A line joining all the places on globe, having same angle of dip or inclination is called isoclinic line.

The magnetic field lines are the lines drawn in a magnetic field along which a north magnetic pole would move. The magnetic field lines are also known as magnetic lines of force.

A line which joins all the places on earth, having zero angle of declination is called agonic line.

A line that joins all the places on earth, having the same angle of declination is called an isogonic line.

Substances which when placed in a magnetising field are strongly magnetised in the direction of the magnetising field are called ferromagnetic substances.

Substances which when placed in a magnetic field are feebly magnetised in the direction of the magnetising field are called paramagnetic substances.

Substances which when placed in a magnetic field are feebly magnetised in a direction opposite to that of the magnetising field are called diamagnetic substances.

Substances which at room temperature retain their ferromagnetic property for a long period of time are called permanent magnets.

\[\vec{E}=\frac{1}{4\pi\varepsilon_0}\frac{Q}{r^2}\hat{r}\]

\[\vec F\] = q(\[\vec E\] + \[\vec v\] × \[\vec B\])

Vector Form: \[\vec F\] = q(\[\vec v\] × \[\vec B\])

Magnitude Form: F = qv B sin θ

Where:

• q = charge on the particle
• v = speed of the particle
• B = magnetic field strength
• θ = angle between \[\vec v\] and \[\vec B\]

Maximum magnetic force (when v ⊥ B): F_max = qv B

B = μ₀​nI

Where:

• μ_0​ = permeability of free space
• n = number of turns per unit length
• I = current

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

Per unit length:

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

Force acts along the line joining the wires

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

\[B=\frac{\mu_0}{4\pi}\cdot\frac{m}{d^3}\]

\[B=\frac{\mu_0}{4\pi}\cdot\frac{2m}{d^3}\]

If we stretch the index finger, middle finger and thumb of the left hand mutually perpendicular to each other such that the index finger points along the direction of the magnetic field and the middle finger along the direction of current (moving charge), then the thumb represents the direction of the force F experienced by the moving charge.

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.

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:

Tangent law states that, if a magnetic field ‘B’ is applied at right angles to the horizontal component of the earth's field B_H, the needle comes to equilibrium at an angle ‘$\theta ’$ to the magnetic meridian such that, tan θ = `B/B_H`.

For ferromagnetic substances above the Curie temperature, the magnetic susceptibility is inversely proportional to (T − T_C), where T_C is the Curie temperature. Mathematically,

On heating beyond the Curie temperature (T_C(iron) = 770 °C), ferromagnetic substances get converted into paramagnetic materials.

The magnetic susceptibility of a paramagnetic material varies inversely with its absolute temperature. Mathematically,

On cooling, paramagnetic substances get converted to ferromagnetic materials at the Curie temperature.

• 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.

• 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 bar magnet behaves like a solenoid
• Both produce similar magnetic field patterns
• Solenoid Relation: M = NIA

• Diamagnetic substances are weakly repelled and have negative susceptibility.
• Paramagnetic substances are weakly attracted and obey Curie law.
• Ferromagnetic substances are strongly attracted and contain domains.
• Ferromagnets become paramagnetic above the Curie temperature.
• The comparison table is the most important revision tool for board preparation.