Problem 44 - Master Resource Book in JEE Main Physics

Question

Two inductors of inductance \(L\) each are connected in series with opposite magnetic fluxes. What is the resultant inductance? (Ignore mutual inductance) (a) Zero (b) \(\underline{L}\) (c) \(2 L\) (d) \(3 L\)

Step-by-Step Solution

Answer

The resultant inductance is zero.

Step 1: Understanding Inductors in Series

When inductors are connected in series, the total or equivalent inductance (L_{ ext{eq}}) is generally the sum of the individual inductances. The formula for inductors in series is given by:\[L_{ ext{eq}} = L_1 + L_2 + ext{...} + L_n\]In this case, each inductor has the same inductance L, so if they were aiding, it would simply be L + L = 2L.

Step 2: Consider Opposite Magnetic Fluxes

Since the inductors are connected such that their magnetic fluxes are opposite, the effective inductance subtracts one from the other. This can be treated as:\[L_{ ext{eq}} = L_1 - L_2\]Given each inductor has inductance L, we substitute to find:\[L_{ ext{eq}} = L - L\]

Step 3: Calculate the Resultant Inductance

Substitute the known values from the previous steps:\[ L_{ ext{eq}} = L - L = 0\]This calculation shows that the resultant inductance of two inductors in series with opposite magnetic fluxes is zero.

Key Concepts

Inductors

Inductors are essential components in electrical circuits, particularly in applications involving alternating current (AC). Their primary function is to store energy in the form of a magnetic field when an electric current flows through them. This ability is defined by a property called inductance, often denoted by the symbol \( L \). Inductance is measured in henrys (H).

Inductors are characterized by their winding of conductive wire, usually copper, coiled around a core material. This construction allows them to form a magnetic field when current passes through, resisting changes in current flow via electromagnetic induction.

In series circuits, inductors add their inductance values together for a total equivalent inductance. This is a straightforward sum if the magnetic fluxes from each inductor are aligned in the same direction. However, when fluxes oppose each other, as in the exercise, calculations become slightly more complex, leading to different results in the total inductance.

Magnetic Flux

Magnetic flux is a key concept related to inductance. It defines the amount of magnetic field passing through a given area, usually the loops of an inductor coil. Represented by the symbol \( \Phi \), magnetic flux is measured in webers (Wb).

When current flows through an inductor, it generates a magnetic field. This field's strength and direction are considered as the magnetic flux. The inductance of an inductor is directly related to its magnetic flux, as higher flux indicates a higher capacity to oppose changes in current. The relationship between induced electromotive force and magnetic flux is given by Faraday's Law of Electromagnetic Induction: \( \varepsilon = - \frac{d\Phi}{dt} \), where \( \varepsilon \) is the induced electromotive force.

Opposing magnetic fluxes occur when the direction of the magnetic fields produced by two inductors aligned in series are opposite. This results in a cancellation effect, drastically affecting their combined inductance, as seen in the exercise.

Resultant Inductance

In circuits with multiple inductors, understanding the concept of resultant or equivalent inductance is crucial. It describes the total inductance experienced by current flowing through the in-series component arrangement. When inductors are in series with aiding magnetic flux, the resultant inductance is simply the sum of the individual inductances. However, this changes when the magnetic fluxes oppose each other.

As illustrated in the exercise, if two inductors, each with an inductance \( L \), have opposing magnetic fluxes, the resultant inductance is calculated by subtracting the opposing flux. This cancellation yields \( L_{eq} = L - L = 0 \).

Such setups can lead to no net inductance, effectively creating a circuit that behaves as though there are no inductors present. This concept is particularly significant in applications where neutralization of inductance is required, such as in certain AC circuit configurations.