Magnetostatics-1 Notes & Topics for GATE EC/EE 2022

In this article, you will find the Study Notes on Magnetostatics-1 which will cover the topics such as Introduction, Biot-savart's Law, Magnetic field due to an infinite and finite conductor, a force due to the Magnetic field.

1. Introduction

Magnetostatics is a branch of electromagnetic studies involving magnetic fields produced by steady non-time varying currents. Evidently, currents are produced by moving charges undergoing translational motion. An effective current (called magnetisation current) is also produced if magnetic dipoles are nonuniformly distributed. The magnetic field B exerts a force perpendicular to the velocity of charged particles. The force
to act on a charge q is

The magnetic field does not do any work on charged particles since the rate of work v.F is independent of B;

As we move from electrostatics to magnetostatics we take into consideration two of Maxwell’s equations and begin using the following equations:

Point or Differential Form:

Integral Form:

Magnetic field: A static magnetic field can be produced by a permanent magnet or a current-carrying conductor. A steady current of I amperes flowing in a straight conductor produces a magnetic field around it. The field exists as concentric circles having centres at the axis of the conductor. If you hold the current-carrying conductor by the right hand so that the thumb points the direction of current flow, then the fingers point the direction of the magnetic field. The unit of magnetic flux is Weber. One Weber equals 10⁸ Maxwell.
Magnetic flux density (B): The magnetic flux per unit area is called magnetic flux density (or) magnetic induction vector. The unit of B is Weber/m2 (or), Tesla. The magnetic flux through any surface is the surface integral of the normal component of B. The magnitude and direction of B due to a current-carrying conductor is given by ‘Biot – savant's law’.

2. Biot-Savart's Law

BIOT and SAVART from their experimental observation deduced a mathematical expression for the elementary magnetic flux density produced by a current element at any particular point of observation (p). According to this law considering a current element of length ‘dl’ carrying a current ‘I’, the magnetic flux density at a point of observation ‘p’ is elementary field intensity.

Magnetic field intensity due to entire conductor can be obtained by line integral,

Curl deals with rotation. The current element vector and distance vector have no rotation. Therefore curl of 'Idl' and curl of 'r' vanish.

This equation is called point form, field form, vector form or differential form of BIOT – SAVART law. It is also called second Maxwell’s equation.

3. Magnetic Field Intensity for Various Current Distribution

Line Current:

Surface Current: Surface current density J_s, measured in (A/m)

Volume Current: Volume current density J, measured in (A/m²)

4. Magnetic field due to an infinite straight conductor

Consider an infinite straight conductor along the z-axis and carrying a current I along with the positive Z-direction. Let ‘p’ be the point of observation on the x-y plane at a distance ‘r’ from the z-axis.

Let 'Idl' = small current element We know that,

Net magnetic field,

5. Magnetic field due to a finite conductor

Let us consider a finite conductor of length MN, for the sake of generality OM ≠ ON. Let ‘p’ be the point of observation on XY plane.

Net magnetic field:

Magnetic field due to infinite conductor, i.e α = 0, β = 180^o:

Magnetic field due to semi infinite conductor, i.e ∝ = 90o, β = 180^o:

Magnetic field due to finite along the perpendicular bisector, i.e α = 180-β:

6. Forces due to Magnetic Field

Forces on a Charged Particle:
- Magnetic Force: An electric field acting on a charge produces a force, so it is with magnetic flux acting on a moving charge
- Electric Force: where

Lorentz Force Equation: For moving charge Q in the presence of both electric and magnetic fields,

Force on a current-carrying conductor: A charged particle in motion receives a force from B, so it follows that B acting on current flow in a wire imparts force to the charged particles in the wire and hence the wire itself,