Revision: Magnetic Effects of Current and Magnetism >> Moving Charges and Magnetism Physics Science (English Medium) Class 12 CBSE

Electromagnetism is the branch of physics that studies the relationship between electric charges, electric current, and magnetic effects.

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

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.

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 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.

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.

Deflection produced per unit current.

Deflection produced per unit voltage.

An instrument used to measure the potential difference between two points in an electrical circuit, always connected in parallel with the component across which the voltage drop is to be measured, is called a voltmeter.

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

\[\vec F\] = 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 \[\vec B\]

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

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:

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

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

Where:

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

F = IL × 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

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

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

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.

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.

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.

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:

Statement

In any closed loop of an electric circuit, the algebraic sum of all changes in potential is zero. 

Derivation

Consider a charge moving around a closed loop. After completing one full loop, the charge returns to its starting point. Since electric potential depends only on position, the net change in potential over a complete loop must be zero. 

Therefore, in a closed loop,

If a loop contains cells and resistors, then the total emf supplied by the sources is equal to the total potential drop across the resistors. Thus,

Conclusion

Kirchhoff's Second Law is a direct consequence of the conservation of energy.

At any junction, the sum of currents entering = the sum of currents leaving.

Example: I₁ + I₃ = I₂ + I₄​. Based on conservation of charge.

The algebraic sum of potential differences in a closed loop is zero.

Based on conservation of energy.

Statement

At any junction in an electric circuit, the sum of currents entering the junction is equal to the sum of currents leaving the junction. 

Derivation

When the current in a circuit is steady, charge does not accumulate at any junction. Therefore, the amount of charge entering the junction per second must be equal to the amount of charge leaving the junction per second. 

If currents I₁​ and I₂ enter a junction and currents I₃​ and I₄​ leave it, then

or

Hence,

Conclusion

Kirchhoff's First Law is a direct consequence of the conservation of charge. 

• Kirchhoff's laws are used for complex circuits. 
• Kirchhoff's First Law: Total current entering a junction = total current leaving a junction. 
• Kirchhoff's Second Law: Total potential rise in a closed loop = total potential drop in the loop. 
• KCL is based on conservation of charge. 
• KVL is based on conservation of energy. 
• Mathematical forms are ∑I = 0 and ∑V = 0. 
• The correct sign convention is essential in numericals.