A solenoid has a radius of 2.00 cm and 1000 turns per...

Key Concepts

Faraday’s Law of Induction

This fundamental principle of electromagnetism states that a changing magnetic flux through a closed loop induces an electromotive force (EMF) in that loop. It is encapsulated by the integral form of Maxwell’s equation, where the line integral of the electric field around a closed path equals the negative rate of change of the magnetic flux enclosed by that path. This concept is crucial when analyzing induced electric fields due to time-varying currents or magnetic fields.

Magnetic Field in a Solenoid

A solenoid is a coil of wire that produces a nearly uniform magnetic field inside when a current passes through it. The magnetic field strength within an ideal solenoid can be expressed as B = ?? n I, where ?? is the permeability of free space, n is the number of turns per unit length, and I is the current. Understanding this relation is key when determining the magnetic flux through a given area of the solenoid, which in turn influences the induced EMF.

Induced Electric Field

When the magnetic flux through a region changes with time, an electric field is induced even in regions where there is no conductor present. This non-conservative induced electric field can be analyzed using the Maxwell-Faraday equation, which relates the circulation of the electric field to the negative time derivative of the magnetic flux. Such fields are essential for understanding electromagnetic induction phenomena across various configurations.

Time-Dependent Current

The time evolution of the current in a system, especially when it follows an exponential or other variable form, directly affects the magnetic field generated and consequently the rate at which the magnetic flux changes. Calculating the time derivative of the current is essential to determine the rate of change of the magnetic field inside a device like a solenoid, which directly impacts the magnitude of the induced electric field.

Transcript

00:01 This given problem we are having a long solenoid first of all radius of this long solenoid is given as r is equal to 2 .00 centimeter and we have to find electric field due to the emf induced as a result of time wearing electric field.

00:41 Current which is passing through the solenoid that is varying with the time so the magnetic field linked with this solenoid will also be varying with the time which will induce an emf in this solenoid due to which we have to find the electric field at a point p whose distance is given as r is equal to 5 .00 centimeter so we consider an imperial loop passing through this observation point so we will find the magnetic flux linked through this loop but the magnetic flux will be only confined within the solenoid now for this solenoid again number of turns per as 1000 per meter the variation of current is given as 3 into e raised to the power 0 .2 t now the line integral of electric field we know a relation between the electric field and the potential an integral of electric field is given as the potential means the emf induced and emf induced is given as using faraday's laws of electromagnetic induction that is the time rate of change of magnetic plugs using the magnitude only so we are ignoring the signs the negative signs here so for this e dot dl over the closed loop it will be equal to electric field as electric field will remain same at all the points over over this circle shown in red.

02:46 This is e into 2xr is equal to d.

02:51 Phi d t means time derivative of the product of magnetic field with the area.

02:57 Here this area is of the solenoid, means yr square.

03:06 And magnetic field here is given by mu not and i.

03:14 So it will be mu not into n into 3 e ratio bar minus, sorry, this is plus 0 .2 p...

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