Understanding Magnetic Effects of Current and Magnetism

Magnetic effects of current and magnetism form a core part of electromagnetism, explaining the interaction between electric currents and magnetic fields. This topic covers the fundamental principles of how moving charges generate magnetic fields, as well as how materials respond to these fields. It is essential for understanding electrical devices, magnetization of materials, and various applications in physics.

Magnetic Field and Oersted’s Experiment

A magnetic field is defined as a region around a moving electric charge or a current-carrying conductor where magnetic effects are observable. Oersted’s experiment demonstrated that electric current creates a magnetic field, influencing a magnetic needle placed nearby.

When the direction of current through a conductor is reversed, the direction of deflection of a magnetic needle placed nearby also reverses. Increasing the current or reducing the distance between the needle and the conductor enhances this deflection.

Biot-Savart Law

Biot-Savart Law provides the expression for the magnetic field produced by a small current element. The magnitude of the field at a point is given by:

$dB = \dfrac{\mu_0}{4\pi} \dfrac{Idl \sin\theta}{r^2}$

Here, $I$ is current, $dl$ is the length element, $r$ is the distance of the point from the element, $\theta$ is the angle between $dl$ and $r$, and $\mu_0$ is the permeability of free space $(4\pi \times 10^{-7}\ \text{Tm/A})$.

In vector form, $d\vec{B} = \dfrac{\mu_0}{4\pi} \dfrac{I(d\vec{l}\times \hat{r})}{r^2}$, where $d\vec{l}$ is in the direction of current flow.

Applications of the Biot-Savart Law are further explained in detail in the Biot-Savart Law resource.

Magnetic Field Due to Different Conductors

The magnetic field at a point due to a straight current-carrying wire or a circular loop can be calculated using the Biot-Savart Law. For an infinitely long wire, the field at a perpendicular distance $d$ is $B = \dfrac{\mu_0 I}{2\pi d}$.

For a circular coil of radius $R$ carrying current $I$, at a distance $x$ along its axis, $B = \dfrac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$. At the center of the coil ($x=0$), this becomes $B = \dfrac{\mu_0 I}{2R}$.

For detailed derivations and variations, resources are available at Magnetic Field From Wire.

Ampere’s Circuital Law

Ampere’s Circuital Law relates the integrated magnetic field around a closed loop to the net current passing through the loop. Mathematically, it is written as:

$\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{\text{encl}}$

This law is especially useful for calculating fields in symmetric situations, such as inside a long solenoid, around a straight wire, or within a toroid.

Magnetic Field Due to a Solenoid and Toroid

A solenoid is a tightly wound helical coil. The magnetic field inside a long solenoid is uniform and parallel to its axis and given by $B = \mu_0 n I$, where $n$ is the number of turns per unit length.

Outside the ideal solenoid, the magnetic field is considered negligible. At the ends, the field reduces to half of its value inside the solenoid.

A toroid is a solenoid bent into a ring shape. The magnetic field inside a toroid at a distance $r$ from the center is $B = \mu_0 n I$, with the field being zero outside and inside the empty central region.

Motion of a Charged Particle in a Magnetic Field

A moving charge in a uniform magnetic field experiences a force perpendicular to both its velocity and the field, given by $F = q\vec{v}\times\vec{B}$. For a perpendicular entry, it describes a circular path of radius $r = \dfrac{mv}{qB}$ and period $T = \dfrac{2\pi m}{qB}$.

If the velocity has components both perpendicular and parallel to the magnetic field, the trajectory becomes a helix, with pitch $p = v_\parallel T$.

Force on a Current-Carrying Conductor in a Magnetic Field

When a current-carrying conductor of length $L$ is placed in a magnetic field, the force on it is $F = I L B \sin\theta$. The direction of this force is determined by Fleming’s left-hand rule.

Interaction Between Parallel Current-Carrying Wires and Definition of Ampere

Two long, parallel current-carrying wires separated by a distance $d$ exert equal and opposite magnetic forces per unit length on each other, given by $f = \dfrac{\mu_0}{2\pi} \dfrac{I_1 I_2}{d}$.

One Ampere is defined as the constant current which, if maintained in two straight parallel conductors placed 1 meter apart in vacuum, produces a force of $2\times 10^{-7}\ \text{N/m}$ between them.

Magnetic Dipole and Bar Magnet

A bar magnet behaves as a magnetic dipole with north and south poles. Its magnetic dipole moment ($\vec{M}$) is a vector from south to north, with magnitude $M = m \times 2l$, where $m$ is pole strength and $2l$ is the magnetic length.

Current loops and solenoids also act as magnetic dipoles, with moment $M = NI A$, where $N$ is number of turns, $I$ is the current, and $A$ is the area.

Torque and Potential Energy of a Magnetic Dipole

A magnetic dipole in an external uniform magnetic field ($\vec{B}$) experiences a torque $\vec{\tau} = \vec{M} \times \vec{B}$ and has potential energy $U = -\vec{M} \cdot \vec{B} = -MB\cos\theta$.

Maximum torque occurs when the angle between $\vec{M}$ and $\vec{B}$ is $90^\circ$, and potential energy is minimum when $\vec{M}$ is aligned with $\vec{B}$.

Moving Coil Galvanometer

A moving coil galvanometer is used to detect and measure small electric currents. The coil, suspended in a radial magnetic field, experiences a torque proportional to the current, producing a measurable deflection. Adding suitable resistances converts it to an ammeter or a voltmeter.

Magnetic Properties of Materials

Magnetic materials are classified as diamagnetic, paramagnetic, or ferromagnetic, depending on their response to an external magnetic field. Diamagnetic materials are weakly repelled, paramagnetic are weakly attracted, and ferromagnetic are strongly attracted by the field.

Type	Main Property
Diamagnetic	Repelled, weak susceptibility
Paramagnetic	Weakly attracted, small positive susceptibility
Ferromagnetic	Strongly attracted, large positive susceptibility

Magnetic susceptibility $(\chi_m)$ measures how easily a material is magnetized, while relative permeability $(\mu_r)$ characterizes how it conducts magnetic flux, with $\mu_r = 1 + \chi_m$.

Magnetisation, Permeability, and Curie’s Law

The intensity of magnetization $(I)$ is the induced magnetic moment per unit volume. Magnetizing field $(H)$ is the external field applied. Magnetic susceptibility $(\chi_m)$ relates $I$ and $H$ by $\chi_m = I / H$, and permeability $(\mu)$ relates magnetic field and induction as $B = \mu H$.

Curie’s Law for paramagnetic substances states that magnetic susceptibility is inversely proportional to temperature, i.e., $\chi \propto \dfrac{1}{T}$, or $\chi = \dfrac{C}{T}$, where $C$ is the Curie constant.

Hysteresis and Magnetic Field Lines

Hysteresis is the lag between magnetic induction $(B)$ and magnetizing field $(H)$ when a ferromagnetic material is magnetized and demagnetized. The B-H curve exhibits coercivity, retentivity, and saturation characteristics. More on this can be found at Hysteresis.

Magnetic field lines are graphical representations of the direction and strength of magnetic fields. They emerge from the north pole and enter the south pole, forming closed loops and never intersecting each other.

Sample Calculation: Magnetic Field at the Center of a Circular Coil

For a circular coil of radius $R$ with current $I$, the magnetic field at the center is $B = \dfrac{\mu_0 I}{2R}$. This formula is foundational for circular current configurations.

Key Points and Study Resources

Comprehensive study resources, including mock tests and formula sheets, are available for targeted practice. Students may access Magnetic Effects Mock Test 1 to review essential concepts for JEE Main.

• Magnetic field forms around current-carrying conductors
• Biot-Savart Law quantifies the field from small elements
• Ampere’s Law applies to symmetric situations
• Solenoids and toroids exhibit uniform internal fields
• Charged particles in a magnetic field move in circles or helices
• Materials show diamagnetic, paramagnetic, or ferromagnetic properties