Magnetic Effects of Current and Magnetism revision notes

Key Concepts & Definitions

Oersted's Discovery

Moving charges or steady currents produce a magnetic field in the surrounding space. A compass needle aligns tangentially to an imaginary circle centered on the current-carrying wire. Reversing the current reverses the deflection.

Magnetic Field (B\mathbf{B}B)

A vector field produced by moving charges or steady currents, capable of exerting force on other moving charges or currents. Its SI unit is Tesla (T) or Weber/m². A smaller non-SI unit is Gauss (1 G=10−4 T1 \, \text{G} = 10^{-4} \, \text{T}1G=10−4T).

Lorentz Force

The total force exerted on a point charge qqq moving with velocity v\mathbf{v}v in the presence of both an electric field E\mathbf{E}E and a magnetic field B\mathbf{B}B is given by F=q[E+v×B]\mathbf{F} = q[\mathbf{E} + \mathbf{v} \times \mathbf{B}]F=q[E+v×B].

Permeability of Free Space (μ0\mu_0μ0​)

A constant of proportionality in magnetism, exactly equal to 4π×10−7 T m A−14\pi \times 10^{-7} \, \text{T m A}^{-1}4π×10−7T m A−1 in SI units. The speed of light relates to it via c=1/μ0ϵ0c = 1/\sqrt{\mu_0 \epsilon_0}c=1/μ0​ϵ0​​.

Ampere (SI Base Unit)

Formally adopted in 1946, 1 Ampere is the value of steady current which, when maintained in two infinitely long, parallel, straight conductors of negligible cross-section placed 1 metre apart in vacuum, produces a force of 2×10−7 N/m2 \times 10^{-7} \, \text{N/m}2×10−7N/m between them.

Magnetic Dipole Moment (m\mathbf{m}m)

For a planar current loop, m=NIA\mathbf{m} = NI\mathbf{A}m=NIA, where NNN is the number of turns, III is current, and A\mathbf{A}A is the area vector pointing in the direction given by the right-hand thumb rule.

Magnetisation (M\mathbf{M}M)

The net magnetic moment per unit volume of a material, M=mnet/V\mathbf{M} = \mathbf{m}_{net}/VM=mnet​/V. Its unit is A m−1\text{A m}^{-1}A m−1.

Magnetic Intensity (H\mathbf{H}H)

The measure of an external magnetising field independent of the material core, defined as H=Bμ0−M\mathbf{H} = \frac{\mathbf{B}}{\mu_0} - \mathbf{M}H=μ0​B​−M.

Magnetic Susceptibility (χ\chiχ)

A dimensionless quantity representing how a magnetic material responds to an external field, M=χH\mathbf{M} = \chi \mathbf{H}M=χH.

Motion of a Charged Particle in a Magnetic Field

When a charge $q$ of mass $m$ moves with velocity $\mathbf{v}$ in a uniform magnetic field $\mathbf{B}$:

• Perpendicular Motion ($\mathbf{v} \perp \mathbf{B}$): The magnetic force acts as a centripetal force ($qvB = mv^2/r$), causing the particle to execute uniform circular motion.
  • Radius: $r = \frac{mv}{qB}$. This can be written in terms of momentum $p$ or kinetic energy $K$ as $r = \frac{p}{qB} = \frac{\sqrt{2mK}}{qB}$.JEE TIPIf particles (like an alpha particle and a proton) enter a B-field with the same kinetic energy, their radius ratio depends purely on $\frac{\sqrt{m}}{q}$.
  • Cyclotron Frequency (Angular): $\omega = \frac{qB}{m}$.JEE TIPThis frequency is completely independent of the particle's speed or orbital radius, which allows cyclotrons to accelerate particles using a fixed-frequency oscillator. However, relativistic mass increases at high speeds break this independence.
  • Time Period: $T = \frac{2\pi m}{qB}$.
• Oblique Motion (Velocity at angle $\theta$ to $\mathbf{B}$): Follows a helical path.
  • The parallel component $v_{\parallel} = v\cos\theta$ is unaffected by the field.
  • The perpendicular component $v_{\perp} = v\sin\theta$ dictates the radius.
  • Pitch of Helix: Linear distance travelled along $\mathbf{B}$ in one rotation. $p = v_{\parallel} T = \frac{2\pi m v\cos\theta}{qB}$.JEE TIPIf a particle moves into a region where magnetic field lines converge, the increasing $B$ causes the radius and pitch to decrease, potentially trapping or reflecting the particle.
• Work Done: Because magnetic force is always perpendicular to velocity, the power $P = \mathbf{F}\cdot\mathbf{v} = 0$. Magnetic fields NEVER change a particle's kinetic energy or speed.

Magnetic Force on Current-Carrying Conductors

• Straight Conductor: $\mathbf{F} = I (\mathbf{l} \times \mathbf{B})$, where $\mathbf{l}$ is a vector representing the length and direction of steady current.
• Arbitrarily Shaped Conductor: $\mathbf{F} = \int I (d\mathbf{l} \times \mathbf{B})$. In a uniform magnetic field, integrating along the wire depends only on the endpoints, giving $\mathbf{F} = I (\mathbf{L}_{eff} \times \mathbf{B})$.JEE TIPAny closed current loop placed in a strictly uniform magnetic field experiences a net translational magnetic force of exactly zero.
• Force Between Parallel Currents: Two infinite parallel wires separated by distance $d$ exert a mutual force per unit length $f = \frac{\mu_0 I_a I_b}{2\pi d}$.
  • Crucial Rule: Parallel currents ATTRACT each other; antiparallel currents REPEL each other.

Magnetic Field due to Currents (Biot-Savart Law)

• Biot-Savart Law: The infinitesimal magnetic field $d\mathbf{B}$ due to a current element $I d\mathbf{l}$ is $d\mathbf{B} = \frac{\mu_0}{4\pi} \frac{I d\mathbf{l} \times \mathbf{r}}{r^3}$.
• Circular Loop (Centre): At $x = 0$, $B = \frac{\mu_0 I}{2R}$.
• Circular Loop (Axis): At an axial distance $x$ from the centre of a loop of radius $R$: $\mathbf{B} = \frac{\mu_0 I R^2}{2(x^2 + R^2)^{3/2}} \hat{i}$.
• Arc of a Circle [JEE Advanced Topic]: For an arc subtending angle $\theta$ (in radians) at the centre, $B = \frac{\mu_0 I}{4\pi R} \theta$.
• Finite Straight Wire [JEE Advanced Topic]: For a finite wire at a perpendicular distance $d$, $B = \frac{\mu_0 I}{4\pi d} (\sin\theta_1 + \sin\theta_2)$, where $\theta_1, \theta_2$ are the angles subtended by the ends of the wire relative to the perpendicular.
• Infinite Straight Wire: $B = \frac{\mu_0 I}{2\pi d}$. Direction is given by the Right-Hand Grip Rule.

Ampere's Circuital Law

• Equation: $\oint \mathbf{B} \cdot d\mathbf{l} = \mu_0 I_{enc}$.
• Sign Convention: Curl the fingers of your right hand in the direction the Amperian loop is integrated ($\oint d\mathbf{l}$). The thumb points in the direction of positive current $I_{enc}$.
• Solid Cylindrical Wire (Radius $a$):
  • Outside ($r > a$): $B = \frac{\mu_0 I}{2\pi r}$.
  • Inside ($r < a$): Assuming uniform current density, the enclosed current is $I_{enc} = I \frac{r^2}{a^2}$. Thus, $B = \frac{\mu_0 I r}{2\pi a^2}$.
• Ideal Solenoid (Long): $B = \mu_0 n I$ inside, where $n = N/L$. Field outside is zero. At the exact ends of a semi-infinite solenoid, $B = \frac{\mu_0 n I}{2}$ [Advanced Addition].
• Toroid [JEE Advanced Topic]: An endless solenoid bent into a circle. $B = \frac{\mu_0 N I}{2\pi r}$ inside the core; zero everywhere outside.

Current Loop as a Magnetic Dipole & Electrostatic Analog

• Torque: A current loop in a uniform field $\mathbf{B}$ experiences torque $\boldsymbol{\tau} = \mathbf{m} \times \mathbf{B}$. It does not experience a net force.
• Potential Energy: $U = -\mathbf{m} \cdot \mathbf{B}$.
• Equilibrium: Stable equilibrium occurs when $\mathbf{m} \parallel \mathbf{B}$ ($\theta = 0^\circ, U_{min} = -mB$). Unstable equilibrium occurs when $\mathbf{m}$ is antiparallel to $\mathbf{B}$ ($\theta = 180^\circ, U_{max} = +mB$).
• Electrostatic Analogy Table: At far distances ($r \gg l$), a magnetic dipole produces fields identical to an electric dipole if mapped via $p \rightarrow m$ and $1/\epsilon_0 \rightarrow \mu_0$:

• JEE TIPCutting a magnet transverse to its length creates two smaller magnets with unchanged pole strength but halved length (dipole moment halves). Cutting longitudinally halves pole strength (dipole moment halves).

Moving Coil Galvanometer (MCG)

• Principle: A coil in a uniform radial magnetic field experiences a torque $\tau = NIAB$. A spring provides restoring torque $\tau = k\phi$. Equating them: $NIAB = k\phi$. (Radial field ensures $\sin\theta = 1$).
• Current Sensitivity ($I_s$): Deflection per unit current, $I_s = \frac{\phi}{I} = \frac{NAB}{k}$.
• Voltage Sensitivity ($V_s$): Deflection per unit voltage, $V_s = \frac{\phi}{V} = \frac{NAB}{kR_G}$.
• Ammeter Conversion: Connect a small shunt $r_s$ in parallel. $r_s = \frac{I_g R_G}{I - I_g}$. Equivalent resistance is very small ($\approx r_s$).
• Voltmeter Conversion: Connect a large resistance $R$ in series. $R = \frac{V}{I_g} - R_G$. Equivalent resistance is very large ($R_G + R$).

Magnetism and Gauss's Law

• Gauss's Law for Magnetism: $\oint \mathbf{B} \cdot d\mathbf{S} = 0$. The net magnetic flux through any closed surface is always zero. This proves isolated magnetic monopoles do not exist.
• Field Lines: Magnetic lines form continuous, closed loops. They never intersect. Inside a bar magnet, lines travel South to North.

Magnetic Properties of Materials

• Diamagnetic: Have negative susceptibility ($-1 \le \chi < 0$), $\mu_r < 1$. Net atomic dipole moment is zero. External B-field induces opposing moments (Lenz's Law). They are repelled by magnets and move from strong to weak B-field regions.
  • Superconductors exhibit the Meissner effect: perfect diamagnetism ($\chi = -1$, $\mu_r = 0$) and perfect conductivity, completely expelling magnetic fields.
• Paramagnetic: Positive, small susceptibility ($0 < \chi < \epsilon$), $\mu_r > 1$. Atoms possess permanent dipole moments randomized by thermal agitation. They are weakly attracted, moving from weak to strong fields. Obeys Curie's Law.
• Ferromagnetic: Huge positive susceptibility ($\chi \gg 1$), $\mu_r \gg 1$. Dipoles spontaneously align in macroscopic volumes called domains ($\approx 1\text{ mm}$ size, containing $\approx 10^{11}$ atoms). External fields cause domains parallel to the field to grow and align.
  • Hard Ferromagnets: Retain magnetisation after field removal (e.g., Alnico). Used for permanent magnets.
  • Soft Ferromagnets: Magnetisation disappears rapidly (e.g., Soft iron). Used in electromagnets/transformers.
• Curie Temperature: At high temperatures, thermal agitation destroys domain structures; ferromagnets transition into paramagnets.

Earth's Magnetism [JEE Advanced Topic]

• The Earth acts as a giant magnetic dipole, with a field strength of $\approx 3.6 \times 10^{-5} \, \text{T}$.
• Declination: The angle between the geographic true north (axis of rotation) and the magnetic north meridian.
• Angle of Dip (Inclination): The angle made by the total Earth's magnetic field vector with the horizontal plane. It is $0^\circ$ at the magnetic equator and $90^\circ$ at the magnetic poles.
• Horizontal Component ($B_H$): $B_H = B_E \cos(\text{Dip})$.

Important Graphs & Diagrams

• $B$ vs $r$ for a Solid Cylindrical Wire: For $r < a$ (inside), the graph is a straight line passing through the origin ($B \propto r$). At the surface $r=a$, it peaks. For $r > a$ (outside), the graph is a rectangular hyperbola ($B \propto 1/r$).
• Hysteresis Loop [JEE Advanced]: Plot of B vs H for ferromagnets. It exhibits hysteresis (lagging). The intercept on the positive B-axis is Retentivity (residual magnetism when $H=0$). The intercept on the negative H-axis is Coercivity (reverse field needed to completely demagnetise the material, $B=0$).

Standard Derivations & Step-by-Step Problem Solving

• Field on Axis of Circular Loop: A current element $Id\mathbf{l}$ at the top of the loop creates $d\mathbf{B}$ angled away from the axis. This vector splits into $dB_x$ (axial) and $dB_\perp$ (perpendicular). The diametrically opposite element creates an equal $dB_\perp$ in the opposite direction. $\int dB_\perp = 0$. We only integrate $dB_x = dB\cos\theta$, leading to the final formula.
• Torque on Rectangular Loop ($A = ab$): If the normal to the loop makes angle $\theta$ with $\mathbf{B}$, forces on arms of length $a$ are zero/collinear. Forces on arms of length $b$ are $F = IbB$. These forces are equal, opposite, and separated by a perpendicular distance $a\sin\theta$. Torque is force $\times$ distance $= (IbB)(a\sin\theta) = I(ab)B\sin\theta = mB\sin\theta$.

Conditions & Limitations

• Lorentz Force Formula: Valid only for point charges. A magnetic force acts strictly if $|\mathbf{v}| \neq 0$ and $\mathbf{v}$ is not completely parallel to $\mathbf{B}$.
• Biot-Savart / Ampere's Law: Strictly applicable only for steady, non-fluctuating (time-independent) currents.
• Solenoid Approximation: $B = \mu_0 n I$ is strictly valid only deep inside the interior of a solenoid where length $\gg$ radius. At the extreme open ends, field halves to $\frac{1}{2}\mu_0 n I$.
• Dipole Approximation: Expressions mapping the magnetic field to electric dipole fields are valid strictly when distance $r$ is much larger than the dipole's physical dimensions ($r \gg l$).

Previous Year JEE Topics

• Helical Motion Calculations: Finding pitch, coordinate positions, or time periods for particles entering at oblique angles.
• Crossed Fields (Velocity Selector): Tuning $\mathbf{E}$ and $\mathbf{B}$ ($v = E/B$) so an ion beam passes undeflected.
• Solid Cylinder Ampere's Law: Testing the internal $B \propto r$ relationship via graphs or numerical ratios.
• Complex Coil Geometries: Applying Biot-Savart to find net $\mathbf{B}$ at the origin from combinations of straight wires and circular arcs.
• Galvanometer Modifications: Analyzing percentage errors introduced when an ammeter (with finite shunt resistance) or voltmeter alters the original circuit current/voltage.
• Dia/Para/Ferromagnetic Distinctions: Identifying materials purely based on whether they seek weaker or stronger magnetic fields in non-uniform setups, or through their $\chi$ values.

TOP 10 MCQ TRAPS

• [JEE TIP] Trap 1 - The Magnetic Kinetic Energy Monopoly:

  • Misconception: A powerful uniform magnetic field performs work on a moving charged particle, thereby scaling up its kinetic energy or absolute speed.
  • Correct Understanding: Because the magnetic Lorentz force is defined by a cross product ($\mathbf{F} = q(\mathbf{v} \times \mathbf{B})$), the force vector is always strictly perpendicular to the instantaneous velocity vector. Consequently, the power ($\mathbf{F} \cdot \mathbf{v}$) and work done are identically zero. A magnetic field can rotate the velocity vector and alter the direction of momentum, but it can never change the kinetic energy or speed of a free particle.

• [JEE TIP] Trap 2 - The Electrostatic Current Analogy Inversion:

  • Misconception: Drawing a false structural parallel to electrostatics (where like charges repel), parallel wires carrying current in the same direction exert a repulsive force on each other.
  • Correct Understanding: The parallel current rule runs exactly opposite to the electrostatic charge rule. Parallel currents flowing in the identical direction attract each other, while anti-parallel currents flowing in opposite directions repel each other. This is verified by applying the right-hand rule to find the magnetic field of the first wire and evaluating the $\mathbf{I}\mathbf{l} \times \mathbf{B}$ force vector on the second.

• [JEE TIP] Trap 3 - Cyclotron Frequency Velocity Independence:

  • Misconception: The orbital time period and cyclotron frequency of a charged particle revolving in a uniform magnetic field scale up as the particle's speed or orbital track radius expands.
  • Correct Understanding: The angular frequency ($\omega = \frac{qB}{m}$) and the total time period ($T = \frac{2\pi m}{qB}$) are completely independent of both the particle's linear velocity and its orbital radius. When a particle accelerates (e.g., in a cyclotron), its path radius expands proportionally ($r \propto v$), ensuring that it completes its larger orbit in the exact same time interval.

• [JEE TIP] Trap 4 - The Galvanometer Sensitivity Scaling Deadlock:

  • Misconception: Doubling the total number of turns in a moving coil galvanometer doubles both its current sensitivity and its voltage sensitivity simultaneously.
  • Correct Understanding: Winding twice the number of turns ($N$) does double the current sensitivity ($\alpha / I = \frac{NBA}{k}$). However, it also requires doubling the physical length of the wire, which doubles the coil's internal electrical resistance ($R$). Because voltage sensitivity is defined as current sensitivity divided by resistance ($V_s = \frac{I_s}{R}$), the scaling factors in the numerator and denominator cancel out, leaving the voltage sensitivity absolutely unchanged.

• [JEE TIP] Trap 5 - The Non-Zero Enclosed Magnetic Flux Fallacy:

  • Misconception: Gauss's Law for magnetism allows for a non-zero net magnetic flux leaking through a closed surface if an isolated, single magnetic pole is enclosed deep inside it.
  • Correct Understanding: The net magnetic flux passing through any closed surface is universally and identically zero ($\oint \mathbf{B} \cdot d\mathbf{S} = 0$). This is a fundamental Maxwell equation proving that magnetic poles always exist as tightly bound dipoles (North and South paired together). Isolated magnetic monopoles do not exist in classical physics; every field line entering a closed surface must also exit it.

• [JEE TIP] Trap 6 - Magnetic Field Line Pole Terminations:

  • Misconception: Similar to electrostatic field lines that start at a positive charge and terminate at a negative charge, magnetic field lines physically originate at the North pole and terminate at the South pole.
  • Correct Understanding: Electrostatic lines are non-closed, but magnetic field lines form continuous, endless closed loops. Externally, they appear to emerge from the North pole and travel toward the South pole. However, to complete the mandatory loop, they travel from the South pole back to the North pole inside the core of the magnet. They possess no starting or ending points.

• [JEE TIP] Trap 7 - Identical Energy Orbital Tracking Ratios:

  • Misconception: If two different particles possessing different masses (such as an alpha particle and a proton) enter a uniform magnetic field with the exact same kinetic energy, their orbital radii are directly proportional to their masses.
  • Correct Understanding: The trajectory radius is governed by momentum ($r = \frac{p}{qB}$). Because momentum relates to kinetic energy ($K$) via the radical expression $p = \sqrt{2mK}$, the radius equation transforms into $r = \frac{\sqrt{2mK}}{qB}$. Therefore, at a constant kinetic energy, the ratio of their tracking radii depends strictly on the specific charge metric $\frac{\sqrt{m}}{q}$, not a linear mass ratio.

• [JEE TIP] Trap 8 - Irregular Closed Loop Translational Force:

  • Misconception: An irregularly shaped or asymmetrical closed current loop placed inside a uniform magnetic field experiences a net translational force pulling it along the field vectors.
  • Correct Understanding: Any closed current-carrying loop placed within a strictly uniform magnetic field experiences a net translational vector force of exactly zero ($\mathbf{F}_{\text{net}} = \mathbf{0}$), regardless of how irregular or contorted its geometry is. This is because the vector sum of its length segments ($\oint d\mathbf{l}$) around any closed loop is zero. The loop may experience an intense net torque ($\boldsymbol{\tau} = \mathbf{M} \times \mathbf{B}$) causing rotation, but it will not translate.

• [JEE TIP] Trap 9 - The Negative Charge Lorentz Direction Drop:

  • Misconception: The vector direction of the magnetic force acting on a moving electron is determined directly by curling your right-hand fingers from the velocity vector $\mathbf{v}$ toward the magnetic field vector $\mathbf{B}$.
  • Correct Understanding: The standard right-hand rule computes exclusively the direction of the raw vector cross product $\mathbf{v} \times \mathbf{B}$. Because an electron carries a negative charge ($q = -e$), the scalar value flips the vector sign in the Lorentz force equation ($\mathbf{F} = -e(\mathbf{v} \times \mathbf{B})$). The actual force vector points exactly $180^\circ$ opposite to the direction predicted by the standard right-hand rule. Failing to invert this vector is a major source of errors in trajectory equations.

• [JEE TIP] Trap 10 - Solid Cylinder Ampere Field Inversion:

  • Misconception: The magnetic field profile mapping the region directly outside a solid cylindrical current-carrying wire follows the identical linear distance relationship as the field deep inside the wire core.
  • Correct Understanding: The field rules invert at the boundary. Deep inside a solid cylindrical wire carrying a uniform current density, Ampere's Law shows that the enclosed current scales with area ($r^2$), making the internal magnetic field directly proportional to the radial distance ($B \propto r$). Outside the wire boundary, the full current is enclosed, and the system behaves like an infinitely thin line current, making the external magnetic field inversely proportional to the distance ($B \propto \frac{1}{r}$).