Induced Electric Field (Time-Varying Magnetic Field)

Electromagnetic induction is a fundamental principle in physics that describes the generation of an electric field due to a changing magnetic field. This phenomenon was discovered by Michael Faraday and is encapsulated in Faraday's Law of Induction.

Faraday's Law of Induction

Faraday's Law states that the induced electromotive force (EMF) in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit. Mathematically, this is expressed as:

$$ \mathcal{E} = -\frac{d\Phi_B}{dt} $$

where $\mathcal{E}$ is the induced EMF and $\Phi_B$ is the magnetic flux, which is defined as:

$$ \Phi_B = \int_S \mathbf{B} \cdot d\mathbf{A} $$

Here, $\mathbf{B}$ is the magnetic field, and $d\mathbf{A}$ is a differential area vector perpendicular to the surface $S$ through which the magnetic field lines pass.

Induced Electric Field

When a magnetic field changes with time, it induces an electric field. Unlike the electric field produced by static charges, which is conservative (path-independent), the induced electric field is non-conservative (path-dependent). This means that the work done in moving a charge around a closed loop in the presence of a time-varying magnetic field is non-zero.

The induced electric field $\mathbf{E}$ can be related to the rate of change of the magnetic field by Maxwell's equation for Faraday's Law:

$$ \nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t} $$

This equation tells us that the curl of the electric field is equal to the negative rate of change of the magnetic field.

Lenz's Law

Lenz's Law gives the direction of the induced EMF and current resulting from electromagnetic induction. It states that the direction of the induced current is such that it opposes the change in magnetic flux that produced it. This is the negative sign in Faraday's Law.

Differences and Important Points

Aspect	Static Electric Field	Induced Electric Field
Source	Static charges	Time-varying magnetic field
Nature	Conservative	Non-conservative
Work Done in a Closed Loop	Zero	Non-zero
Governed by	Coulomb's Law	Faraday's Law and Lenz's Law
Mathematical Description	$\nabla \times \mathbf{E} = \mathbf{0}$	$\nabla \times \mathbf{E} = -\frac{\partial \mathbf{B}}{\partial t}$

Examples

Example 1: Solenoid with Time-Varying Current

Consider a solenoid with a time-varying current $I(t)$. This current produces a magnetic field inside the solenoid that changes with time, which in turn induces an electric field.

The magnetic flux through a loop surrounding the solenoid can be expressed as:

$$ \Phi_B = B \cdot A = \mu_0 n I(t) A $$

where $n$ is the number of turns per unit length of the solenoid, $A$ is the cross-sectional area, and $\mu_0$ is the permeability of free space.

Using Faraday's Law, the induced EMF is:

$$ \mathcal{E} = -\frac{d\Phi_B}{dt} = -\mu_0 n A \frac{dI}{dt} $$

Example 2: Rotating Loop in a Magnetic Field

A rectangular loop of wire with sides $l$ and $w$ rotates with an angular velocity $\omega$ in a uniform magnetic field $B$ that is perpendicular to the axis of rotation. The magnetic flux through the loop changes as a function of time due to the rotation.

The magnetic flux at any time $t$ is given by:

$$ \Phi_B = B \cdot lw \cdot \cos(\omega t) $$

The induced EMF in the loop is:

$$ \mathcal{E} = -\frac{d\Phi_B}{dt} = B \cdot lw \cdot \omega \cdot \sin(\omega t) $$

The induced electric field creates a current in the loop, which can be calculated using Ohm's Law if the resistance of the loop is known.

Conclusion

The concept of an induced electric field due to a time-varying magnetic field is a cornerstone of electromagnetism and has numerous applications in technology, such as in the operation of generators, transformers, and inductive charging devices. Understanding Faraday's Law, Lenz's Law, and the non-conservative nature of the induced electric field is essential for studying and applying electromagnetic induction.