Differential Equations In Electromagnetism: Solving Problems With Calculus

Edited By Ramraj Saini | Updated on Aug 11, 2023 03:30 PM IST

In this world, where electricity and magnetism shape our everyday lives – from the light bulbs that brighten up our rooms to smartphones, we can't live without them. Have you ever wondered how these forces interact and create the marvels of our modern world? Well, that's where electromagnetism comes into play!

Differential Equations In Electromagnetism: Solving Problems With Calculus

Now, let's talk about "Differential Equations." Don't let the name scare you – think of them as magical rules that help us understand how things change over time. Just like a recipe guides you step by step to bake a delicious cake, differential equations guide us through the changes happening in the world of electromagnetism. They are like superhero equations that can predict how electric and magnetic fields evolve and interact in various situations.

But how do we use these superhero equations? That's where calculus enters the scene. Calculus is like a toolkit filled with special techniques that allow us to analyse how things change. Whether it's finding the rate of change of an electric field or understanding how a magnetic field varies in space, calculus has got our back.

In this learning journey, we will deep dive into the mysteries of electromagnetism by solving real-world problems using these superhero equations and calculus. We will discover how to predict the behaviour of electric and magnetic fields, and how mastering them can be a pivotal tool for excelling in various competitive exams such as CBSE, JEE Main, JEE Advanced, NEET, and BITSAT.

Electromagnetism And Differential Equations

Electromagnetism is the study of the interplay between electric and magnetic fields, elucidated by the remarkable equations formulated by James Clerk Maxwell. The two main components of these equations are Gauss's law for electricity, Gauss's law for magnetism, Faraday's law of electromagnetic induction, and Ampère's law with Maxwell's addition. These equations are a quintessential part of the electromagnetic theory and are presented in both integral and differential forms. It's the differential form that brings the magic of calculus into the picture.

Differential equations in electromagnetism allow us to analyse complex systems, predict behaviours of electric and magnetic fields, and solve practical problems. They enable us to model how electric and magnetic fields change over time, given certain conditions.

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The Role of Differential Equations

Differential equations come into play when describing how electric and magnetic fields evolve in different scenarios. For instance, consider a situation where a changing magnetic field induces an electromotive force (EMF) in a conducting loop. This phenomenon is described by Faraday's law of electromagnetic induction, which can be expressed as a differential equation:

∇×E = −∂B/∂t

Here, E represents the electric field, B is the magnetic field, and ∂B/∂t denotes the change in magnetic field with respect to time. This differential equation illustrates how changes in the magnetic field induce a circulating electric field around it.

Solving Problems With Differential Equations

To illustrate the power of using differential equations in electromagnetism, let's consider a practical example involving a changing magnetic field and its induced electric field.

Problem:

A circular loop with radius r is placed in a region where the magnetic field changes with time according to B(t) = αt² + βt, where α and β and are constants. Find the induced electric field E in the loop.

Solution:

To find the induced electric field E in the circular loop, we can use Faraday's law of electromagnetic induction. According to Faraday's law, the induced electromotive force (EMF) around a closed loop is equal to the negative rate of change of magnetic flux through the loop. Mathematically, this can be written as:

EMF = -dϕ/dt, where ϕ is the magnetic flux through the loop.

Magnetic field is given as B(t) = αt² + βt.

The magnetic flux through the circular loop is given by the product of the magnetic field B passing through the loop and the area A of the loop:

ϕ = B⋅A

For a circular loop of radius r, area A = πr2

ϕ = πr² (αt² + βt)

Now, let's differentiate the magnetic flux with respect to time t to find the rate of change of magnetic flux:

dϕ/dt = πr²(2αt + β )

EMF = - dϕ/dt = -πr² (2αt + β)

The Induced electric field E in the loop is related to the induced EMF by the equation.

EMF = E.2πr, where 2πr is the circumference of the loop.

So, we have:

E.2πr = -πr² (2αt + β)

E = -(r/2) (2αt + β)

This is an induced electric field due to changing magnetic fields.

Problem 2 (JEE Main 2019)

A very long solenoid of radius R is carrying current I(t) = kte^-αt where (K>0) as a function of time (t>0). Counterclockwise current is taken to be positive. A circular conducting coil of radius 2R is placed in an equatorial plane of the solenoid and concentric with the solenoid. The current induced in the outer coil is correctly depicted, as a function of time, by:

Solution

Faraday's electromagnetic induction helps us to calculate the induced electric field.

Induced Current

$I= \frac{\varepsilon }{R}=\frac{-N}{R}\frac{d\phi }{dt}$

Wherein

$R\rightarrow$ Resistance

$\frac{d\phi }{dt}\rightarrow$ Rate of change of flux

Given,

$I(t)=kte^{-\alpha t}$

$\phi=(\mu_0NI)A=\mu_0NKte^{-\alpha t}A$

Induced current

$I_(indused)=-\frac{d\phi}{dt}\times\frac{1}{R}$

$=-\mu_0NK[e^{-\alpha t}-\alpha t e^{-\alpha t}]$

$=-Ce^{-\alpha t}(1-\alpha t)$

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Hence, option a) is correct.

Mastering Differential Equations in Electromagnetism

Differential equations form the backbone of advanced electromagnetism problems. Mastering these equations can unlock the ability to understand and predict various electromagnetic phenomena, from electromagnetic waves to the behaviour of circuits. Moreover, a solid foundation in solving differential equations will greatly aid in tackling challenging physics problems in competitive exams.

For aspiring candidates of CBSE, JEE Main, JEE Advanced, NEET, and BITSAT, here are some key takeaways:

Conceptual Clarity: Understand the physical significance of each term in the differential equations. Visualise how changes in fields lead to induced effects.

Mathematical Proficiency: Sharpen your calculus skills. Practice differentiation and integration techniques, as well as solving differential equations of varying complexities.

Problem Solving: Regular practice of problems involving differential equations will enhance your problem-solving skills and boost your confidence.

Application to Real-world Scenarios: Relate differential equations to real-world scenarios. This will help you appreciate the practicality of these equations and their role in technological advancements.

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Now you have a clear idea about electromagnetism, differential equations, and calculus. This will help you not only in competitive exams but also in understanding world events.