JEE Physics: Magnetic Effect of Current Complete Guide

The magnetic effect of current is one of the fundamental concepts in electromagnetism covered extensively in JEE Physics. It bridges electricity and magnetism, explaining how moving charges produce magnetic fields, and how these fields interact with currents and charges. This guide covers key laws, concepts, formulas, and problem-solving strategies to master this important topic.

1. Introduction to Magnetic Effects of Current

When electric current flows through a conductor, it produces a magnetic field around it. This was first discovered by Hans Christian Ørsted. The magnetic field can exert force on other current-carrying wires and moving charges. Understanding this interaction is critical for many devices such as motors, generators, electromagnets, and transformers.

2. Magnetic Field and Its Properties

Magnetic field \(\vec{B}\) at a point is a vector quantity representing the magnetic influence of electric currents and magnetic materials. It is measured in Tesla (T) in SI units.

3. Biot-Savart Law

Biot-Savart law gives the magnetic field produced at a point by a small element of current-carrying conductor:

$$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$$

Where:

Biot-Savart law is fundamental but often used for calculating fields in symmetrical current configurations.

3.1 Magnetic Field on the Axis of a Circular Loop

For a circular loop of radius \(R\) carrying current \(I\), magnetic field at the center is:

$$B = \frac{\mu_0 I}{2R}$$

At a point along the axis at distance \(x\) from the center:

$$B = \frac{\mu_0 I R^2}{2 (R^2 + x^2)^{3/2}}$$

3.2 Magnetic Field on the Axis of a Circular Arc

For an arc subtending angle \(\theta\) at the center:

$$B = \frac{\mu_0 I \theta}{4 \pi R}$$

4. Ampere’s Circuital Law

Ampere's law states:

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

The line integral of magnetic field around a closed path equals \(\mu_0\) times the net current enclosed by the path. It is especially useful for high symmetry cases.

4.1 Magnetic Field due to a Long Straight Current-Carrying Wire

Using Ampere’s law, magnetic field at distance \(r\) from a long straight wire carrying current \(I\):

$$B = \frac{\mu_0 I}{2\pi r}$$

The field lines form concentric circles around the wire.

4.2 Magnetic Field Inside a Solenoid

For a long solenoid with \(n\) turns per unit length carrying current \(I\):

$$B = \mu_0 n I$$

Inside the solenoid, the field is uniform and along the axis; outside it is nearly zero.

4.3 Magnetic Field Inside a Toroid

For a toroid with mean radius \(r\) and \(N\) turns carrying current \(I\):

$$B = \frac{\mu_0 N I}{2\pi r}$$

5. Force on Current-Carrying Conductors

A conductor carrying current \(I\) placed in a magnetic field \(\vec{B}\) experiences a force given by:

$$\vec{F} = I \vec{L} \times \vec{B}$$

Where \(\vec{L}\) is the length vector of the conductor segment in the direction of current. The magnitude of force:

$$F = I L B \sin \theta$$

\(\theta\) is the angle between \(\vec{L}\) and \(\vec{B}\).

5.1 Fleming’s Left Hand Rule

To find direction of force on conductor:

6. Force Between Two Parallel Current-Carrying Wires

Two parallel wires separated by distance \(d\) carrying currents \(I_1\) and \(I_2\) exert force per unit length on each other:

$$\frac{F}{L} = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d}$$

Force is attractive if currents are in the same direction and repulsive if opposite.

7. Torque on a Current-Carrying Loop in Magnetic Field

A rectangular current loop of area \(A\) carrying current \(I\) placed in a magnetic field \(\vec{B}\) experiences torque:

$$\tau = I A B \sin \theta$$

\(\theta\) is angle between normal to loop and \(\vec{B}\). This principle is the basis for electric motors.

8. Moving Charges in Magnetic Field

A charge \(q\) moving with velocity \(\vec{v}\) in magnetic field \(\vec{B}\) experiences Lorentz force:

$$\vec{F} = q \vec{v} \times \vec{B}$$

Magnitude:

$$F = q v B \sin \theta$$

This force causes circular or helical motion depending on velocity components.

8.1 Radius of Circular Path

For velocity perpendicular to \(\vec{B}\):

$$r = \frac{m v}{q B}$$

Where \(m\) is mass of the particle.

8.2 Cyclotron Frequency

$$f = \frac{q B}{2 \pi m}$$

Frequency of revolution is independent of speed.

9. Magnetic Dipole and Magnetic Moment

A current loop behaves like a magnetic dipole with magnetic moment:

$$\vec{\mu} = I \vec{A}$$

Torque on magnetic dipole in field \(\vec{B}\):

$$\vec{\tau} = \vec{\mu} \times \vec{B}$$

10. Moving Conductor in Magnetic Field and Induced EMF

A conductor moving with velocity \(v\) perpendicular to magnetic field \(B\) induces emf:

$$\mathcal{E} = B L v$$

This is the basis of electromagnetic induction.

11. Summary of Important Formulas

Concept Formula Notes
Biot-Savart Law $$d\vec{B} = \frac{\mu_0}{4\pi} \frac{I d\vec{l} \times \hat{r}}{r^2}$$ Field from current element
Field by Long Wire $$B = \frac{\mu_0 I}{2\pi r}$$ At distance \(r\)
Field at Loop Center $$B = \frac{\mu_0 I}{2R}$$ Loop radius \(R\)
Force on Wire $$\vec{F} = I \vec{L} \times \vec{B}$$ Length \(L\) in field
Force Between Wires $$\frac{F}{L} = \frac{\mu_0}{2\pi} \frac{I_1 I_2}{d}$$ Distance \(d\)
Torque on Loop $$\tau = I A B \sin \theta$$ Area \(A\), angle \(\theta\)
Force on Moving Charge $$\vec{F} = q \vec{v} \times \vec{B}$$ Charge \(q\), velocity \(\vec{v}\)

12. Practice Problems

Problem 1:

Find the magnetic field at the center of a circular loop of radius 10 cm carrying current 5 A.

Solution:

$$B = \frac{\mu_0 I}{2 R} = \frac{4 \pi \times 10^{-7} \times 5}{2 \times 0.1} = 3.14 \times 10^{-5} \, \mathrm{T}$$

Problem 2:

Two parallel wires 50 cm apart carry currents 3 A and 5 A in the same direction. Find force per meter between them.

Solution:

$$\frac{F}{L} = \frac{\mu_0}{2 \pi} \frac{I_1 I_2}{d} = \frac{4 \pi \times 10^{-7}}{2 \pi} \frac{3 \times 5}{0.5} = 6 \times 10^{-6} \, \mathrm{N/m}$$