Magnetic Field due to Current

Understanding the magnetic field generated by electric currents is fundamental to electromagnetism. This section explores the laws and principles that govern this phenomenon, crucial for mastering topics in competitive exams like JEE.

Biot-Savart Law

The Biot-Savart Law quantifies the magnetic field produced by a steady electric current. It states that the magnetic field ( $dB$ ) at a point due to a small current element ( $Idl$ ) is directly proportional to the current, the length of the element, and the sine of the angle between the element and the position vector to the point, and inversely proportional to the square of the distance from the element to the point.

The Biot-Savart Law describes how a tiny segment of a current-carrying wire creates a magnetic field.

The law provides a mathematical formula to calculate the magnetic field contribution from each infinitesimal part of a wire. This contribution depends on the current, the length of the segment, its orientation, and the distance to the observation point.

Mathematically, the Biot-Savart Law is expressed as: $dB = \frac{\mu_0}{4\pi} \frac{I dl \sin\theta}{r^2}$ , where $dB$ is the magnetic field strength, $\mu_0$ is the permeability of free space, $I$ is the current, $dl$ is the length of the current element, $\theta$ is the angle between $dl$ and the position vector $r$ , and $r$ is the distance from the current element to the point where the field is being measured. The direction of $dB$ is perpendicular to both $dl$ and $r$ , given by the right-hand rule.

Applications of Biot-Savart Law

By integrating the Biot-Savart Law over the entire length of a current-carrying conductor, we can determine the magnetic field for various geometries.

Conductor Shape	Magnetic Field at Center	Magnetic Field on Axis
Circular Loop (radius R, current I)	$\frac{\mu_0 I}{2R}$	$\frac{\mu_0 I R^2}{2(R^2 + x^2)^{3/2}}$
Straight Wire (infinite length, current I)	$\frac{\mu_0 I}{2\pi r}$ (at distance r)	N/A
Straight Wire (finite length, current I)	$\frac{\mu_0 I}{4\pi r} (\sin\theta_1 + \sin\theta_2)$	N/A

Ampere's Circuital Law

Ampere's Circuital Law provides a simpler way to calculate magnetic fields in situations with high symmetry. It relates the magnetic field around a closed loop to the total current passing through the surface enclosed by the loop.

Ampere's Law relates magnetic fields to enclosed currents in symmetric situations.

This law states that the line integral of the magnetic field around any closed loop is proportional to the total electric current enclosed by the loop. It's a powerful tool for calculating magnetic fields when the current distribution has symmetry.

Mathematically, Ampere's Law is stated as: $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enc}$ , where $\oint \vec{B} \cdot d\vec{l}$ is the line integral of the magnetic field $\vec{B}$ around a closed path, and $I_{enc}$ is the net current enclosed by that path. This law is particularly useful for calculating magnetic fields around infinitely long straight wires, solenoids, and toroids.

Think of Ampere's Law as a shortcut for calculating magnetic fields when the problem has a circular or linear symmetry, much like Gauss's Law simplifies electric field calculations for symmetric charge distributions.

Magnetic Field of Solenoids and Toroids

Solenoids and toroids are common configurations used to generate uniform magnetic fields. Their magnetic field calculations are direct applications of Ampere's Law.

A solenoid is essentially a coil of wire wound into a helix. Inside a long solenoid, the magnetic field is nearly uniform and parallel to the axis. Outside, it is very weak. For a solenoid with $n$ turns per unit length carrying current $I$ , the magnetic field inside is $B = \mu_0 n I$ . A toroid is a doughnut-shaped coil. The magnetic field inside the toroid is tangential to the circular path and its magnitude depends on the distance from the center of the toroid. For a toroid with $N$ turns and average radius $R$ , carrying current $I$ , the magnetic field at a radius $r$ is $B = \frac{\mu_0 N I}{2\pi r}$ .

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What is the key difference in the application of Biot-Savart Law versus Ampere's Circuital Law for calculating magnetic fields?

Biot-Savart Law is a general law applicable to any current distribution, while Ampere's Law is simpler and more efficient for calculating magnetic fields in situations with high symmetry (e.g., straight wires, solenoids, toroids).

Force on a Current-Carrying Wire in a Magnetic Field

A current-carrying wire placed in an external magnetic field experiences a force. This force is a consequence of the magnetic forces acting on the moving charges within the wire.

Current-carrying wires experience a force when placed in a magnetic field.

The force on a straight wire of length $L$ carrying current $I$ in a uniform magnetic field $B$ is given by $F = I L B \sin\theta$ , where $\theta$ is the angle between the direction of the current and the magnetic field. The direction of the force is perpendicular to both the wire and the magnetic field, determined by the right-hand rule.

The force on a current element $Id\vec{l}$ in a magnetic field $\vec{B}$ is given by $d\vec{F} = I d\vec{l} \times \vec{B}$ . For a straight wire of length $L$ in a uniform magnetic field, this integrates to $\vec{F} = I \vec{L} \times \vec{B}$ , where $\vec{L}$ is a vector along the wire in the direction of the current with magnitude equal to the length of the wire. The magnitude of this force is $F = I L B \sin\theta$ .

When is the force on a current-carrying wire in a magnetic field maximum?

The force is maximum when the wire is perpendicular to the magnetic field (i.e., $\theta = 90^\circ$ , $\sin\theta = 1$ ).

Learning Resources

Biot-Savart Law - Physics Classroom(documentation)

Provides a clear explanation of the Biot-Savart Law, its formula, and its applications in calculating magnetic fields.

Ampere's Law - Khan Academy(video)

A video tutorial explaining Ampere's Law, its derivation, and its use in solving problems involving magnetic fields.

Magnetic Field of a Solenoid - HyperPhysics(documentation)

Details the magnetic field inside a solenoid, including the formula and conditions for uniformity.

Force on a Current-Carrying Wire in a Magnetic Field - Physics LibreTexts(documentation)

Explains the force experienced by a current-carrying wire in a magnetic field, including the relevant formula and direction.

JEE Physics - Electromagnetism: Magnetic Field due to Current(video)

A comprehensive video lecture covering magnetic fields due to currents, suitable for JEE preparation.

Magnetic Field of a Toroid - Physics Stack Exchange(blog)

A discussion and explanation of the magnetic field produced by a toroid, often with helpful diagrams.

Biot-Savart Law and Ampere's Law Comparison(blog)

Compares and contrasts Biot-Savart Law and Ampere's Law, highlighting their respective uses and limitations.

JEE Main 2024 Physics: Current Electricity and Magnetism(documentation)

An overview of the Current Electricity and Magnetism syllabus for JEE Main, with links to specific topics.

Magnetic Field Calculations - Brilliant.org(documentation)

Covers various methods for calculating magnetic fields, including Biot-Savart and Ampere's Law, with interactive examples.

NCERT Physics Class 12 - Chapter 4: Moving Charges and Magnetism(documentation)

The official NCERT textbook chapter covering moving charges and magnetism, essential for JEE preparation.