Magnetostatics Notes | EduRev

Electricity & Magnetism

IIT JAM : Magnetostatics Notes | EduRev

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Magnetic Force on Current 

The magnetic field at any point due to steady current is called a magnetostatic field. The magnetic force on a charge Q , moving with velocity v in a magnetic field Magnetostatics Notes | EduRevis, Magnetostatics Notes | EduRevThis is known as Lorentz force law. 

In the presence of both electric and magnetic fields, the net force on Q would be:  
Magnetostatics Notes | EduRev

Current in a Wire

Magnetostatics Notes | EduRev

A line charge λ traveling down a wire at a speed Magnetostatics Notes | EduRev constitutes a current Magnetostatics Notes | EduRev 

Magnetic force on a segment of current-carrying wire is, 

Magnetostatics Notes | EduRev

since Magnetostatics Notes | EduRev  points in the same direction
Magnetostatics Notes | EduRev
Surface Current Density

Magnetostatics Notes | EduRev

When charge flows over a surface, we describe it by the surface current Magnetostatics Notes | EduRev

Magnetostatics Notes | EduRev is the current per unit width-perpendicular to flow.
Also Magnetostatics Notes | EduRev where is σ surface charge density and Magnetostatics Notes | EduRev is its velocity.

Magnetic force on surface current Magnetostatics Notes | EduRev

Volume Current Density

Magnetostatics Notes | EduRev

When the flow of charge is distributed throughout a three-dimensional region, we describe it by the volume current density Magnetostatics Notes | EduRev .

Magnetostatics Notes | EduRev is the current per unit area-perpendicular to flow. 

Also Magnetostatics Notes | EduRev where ρ is volume charge density and Magnetostatics Notes | EduRev is its velocity.

Magnetic force on volume current
Magnetostatics Notes | EduRev
Current crossing a surface S is Magnetostatics Notes | EduRev
Example 1: A wire ABCDEF (with each of side of length L ) bent as shown in figure and carrying a current I is placed in a uniform magnetic induction B parallel to the positive y-direction. Find the force experienced by the wire. 

Magnetostatics Notes | EduRev

FE and BA are parallel to magnetic induction Magnetostatics Notes | EduRev. Magnetic force on each of  them will be zero. DE and CB are perpendicular to Magnetostatics Notes | EduRev. They carry currents in opposite directions forces on them will be equal in magnitude and opposite in direction. The net force due to these portions of wire will be zero. Now force on side DC is Magnetostatics Notes | EduRev


Example 2: A semi–circular wire of radius R carries a current I and is placed in a uniform field B acting perpendicular to the plane of the semi–circle. Calculate force acting on the wire. 


Magnetostatics Notes | EduRevConsider an element of length dl of the wire. The Magnetostatics Notes | EduRev force on this element is obtained by
Magnetostatics Notes | EduRev
 (Horizontal component cancels only perpendicular component add up).


Continuity Equation 

The total charge per unit time leaving a volume V is Magnetostatics Notes | EduRev
Because charge is conserved, whatever flows out through the surface must come at the expense of that remaining inside:
Magnetostatics Notes | EduRev
(The minus sign reflects the fact that an outward flow decreases the charge left in V.) Since this applies to any volume, we conclude that
Magnetostatics Notes | EduRev

This is the precise mathematical statements of local charge conservation. 

Note: When a steady current flows in a wire, its magnitude I must be the same all along the line; otherwise, charge would be piling up somewhere, and it wouldn't be a steady current. Thus for magnetostatic fields Magnetostatics Notes | EduRev and hence the continuity equation becomes:
Magnetostatics Notes | EduRev

Biot-Savart Law

Magnetostatics Notes | EduRev

The magnetic field of a steady line current is given by
Magnetostatics Notes | EduRev

where
Magnetostatics Notes | EduRev
For surface and volume current Biot-Savart law becomes:  
Magnetostatics Notes | EduRev
Magnetic Field due to Wire

Let us find the magnetic field a distance d from a long straight wire carrying a steady current I.  

Magnetostatics Notes | EduRev

In the diagram, Magnetostatics Notes | EduRev points out of the page and has magnitude dl′ sinα = dl′cos θ
Since,
Magnetostatics Notes | EduRev
From Biot–Savart law:
Magnetostatics Notes | EduRev
For Infinite wire:
Magnetostatics Notes | EduRev
Note:

  1. Magnetic field a distance r from a long straight wire carrying a steady current I is
    Magnetostatics Notes | EduRev 
  2. Magnetic field a distance r from a infinite wire carrying a steady current I is:
    Magnetostatics Notes | EduRev
  3. Force (per unit length) of attraction between two long, parallel wires a distance d apart, carrying currents I1 and I2 in same direction are:
    Magnetostatics Notes | EduRev
  4. If currents are in opposite direction they will repel with same magnitude.

Magnetic Field due to Solenoid and Toroid

The magnetic field of a very long solenoid, consisting of n closely wound turns per unit length of a cylinder of radius R and carrying a steady current I is:

Magnetostatics Notes | EduRev 

Magnetic field due to Toroid is
Magnetostatics Notes | EduRev

where N is the total number of turns. 


Example 3: Find the force of attraction between two long, parallel wires a distance d apart, carrying current I1 and I2 in the same direction.

Magnetostatics Notes | EduRev

The field at (2) due to (1) is Magnetostatics Notes | EduRev

Force on (2) is
Magnetostatics Notes | EduRev
Force per unit length is towards (1) and net force is attractive.


Example 4: Find the magnetic field a distance d above the center of a circular loop of radius R, which carries a steady current I.

Magnetostatics Notes | EduRev

The field Magnetostatics Notes | EduRev attributable to the segment Magnetostatics Notes | EduRev as shown. As we integrate Magnetostatics Notes | EduRev 

around the loop, Magnetostatics Notes | EduRev sweeps out a cone. The horizontal components cancel, and the vertical components combine to give.

Magnetostatics Notes | EduRev

Magnetostatics Notes | EduRev


Example 5: Find the force on a square loop placed as shown in figure, near an infinite straight wire. Both the loop and the wire carry a steady current I.
Magnetostatics Notes | EduRev

The force on the two sides cancels.
At the bottom,
Magnetostatics Notes | EduRev 
At the top,
Magnetostatics Notes | EduRev
Thus
Magnetostatics Notes | EduRev


Ampere's Law
Magnetostatics Notes | EduRev

The magnetic field of an infinite wire is shown in the figure (the current is coming out of the page). Let us find the integral of  Magnetostatics Notes | EduRev around a circular path of radius r, centered at the wire, is
Magnetostatics Notes | EduRev

Notice that the answer is independent of r; that is because B decreases at the same rate as the circumference increases. If we use cylindrical coordinates ( r ,φ,z ) , with the current flowing along the z axis,
Magnetostatics Notes | EduRev
In general we can write
Magnetostatics Notes | EduRev
where Ienc is the total current enclosed by the amperian loop.
since
Magnetostatics Notes | EduRev 

Right hand Rule

If the fingers of your right hand indicate the direction of integration around the boundary, then your thumb defines the direction of a positive current.

Magnetostatics Notes | EduRev

Example 6: A steady current I flow down a long cylindrical wire of radius a. Find the magnetic field, both inside and outside the wire, if

  1. The current is uniformly distributed over the outside surface of the wire.
  2. The current is distributed in such a way that J is proportion to r, the distance from the axis.

Magnetostatics Notes | EduRev 

(1)
Magnetostatics Notes | EduRev
(2)
Magnetostatics Notes | EduRev
Magnetostatics Notes | EduRev


Example 7: Find the magnetic field of an infinite uniform surface current Magnetostatics Notes | EduRev , flowing over the x–y plane.

Magnetostatics Notes | EduRev


Since Magnetostatics Notes | EduRev have no x-component because B is ⊥r to x-component i.e. in the direction of Magnetostatics Notes | EduRev
Also, Magnetostatics Notes | EduRev have no z-component:   For y > 0 , B is along Magnetostatics Notes | EduRev and for y < 0 , B is along −Magnetostatics Notes | EduRev thus field cancels each other.

Magnetostatics Notes | EduRev has only y-component: 

For z >0 , Magnetostatics Notes | EduRev points left (Magnetostatics Notes | EduRev) and for z <0 , Magnetostatics Notes | EduRev  point’s right (Magnetostatics Notes | EduRev ).

Draw a rectangular amperian loop parallel to the yz plane and extending an equal distance above and below the surface. Now apply ampere’s law, we find
Magnetostatics Notes | EduRev {One Bl from top segment, and the other from bottom}
Magnetostatics Notes | EduRev

Note: The field is independent of the distance from the plane, just like the electric field  of a uniform surface charge.

Magnetic Vector Potential Magnetostatics Notes | EduRev
Since
Magnetostatics Notes | EduRev
For magnetostatic fields,
Magnetostatics Notes | EduRev
if Magnetostatics Notes | EduRev goes to zero at infinity, Magnetostatics Notes | EduRev for volume current.
For line and surface currents,
Magnetostatics Notes | EduRev


Example 8: What current density would produce the vector potential Magnetostatics Notes | EduRev(where K is   a constant), in cylindrical coordinates ?

Magnetostatics Notes | EduRev 
Magnetostatics Notes | EduRev


Magnetostatic Boundary Condition (Boundary is sheet of current, Magnetostatics Notes | EduRev) 

Just as the electric field suffers a discontinuity at a surface charge, so the magnetic field is discontinuous at a surface current. Only this time it is the tangential component that changes.
Magnetostatics Notes | EduRev

Since
Magnetostatics Notes | EduRev
For tangential components
Magnetostatics Notes | EduRev
Thus the component of Magnetostatics Notes | EduRev that is parallel to the surface but perpendicular to the current is discontinuous in the amount μ0K . A similar amperian loop running parallel to the current reveals that the parallel component is continuous. The result can be summarized in a single formula:
Magnetostatics Notes | EduRev
where Magnetostatics Notes | EduRev is a unit vector perpendicular to the surface, pointing “upward”. Like the scalar potential in electrostatics, the vector potential is continuous across, a boundary:
Magnetostatics Notes | EduRev
For Magnetostatics Notes | EduRev guarantees that the normal component is continuous, and 

Magnetostatics Notes | EduRev in the form
Magnetostatics Notes | EduRev
But the derivative of Magnetostatics Notes | EduRev inherits the discontinuity of Magnetostatics Notes | EduRev 
Magnetostatics Notes | EduRev


Example 9: (a) Find the magnetic vector potential at a distance r from an infinite straight wire carrying a current I. 
(b) Find the magnetic potential inside the wire, if it has radius R and the current is uniformly distributed.

(a)  Magnetostatics Notes | EduRev point in the same direction as I and is a function of r (the distance from  the wire). In cylindrical coordinates
Magnetostatics Notes | EduRev
Magnetostatics Notes | EduRev
(b)
Magnetostatics Notes | EduRev
Magnetostatics Notes | EduRev
where b is arbitrary constant.
Magnetostatics Notes | EduRev must be continuous at R,  Magnetostatics Notes | EduRev 

which means that we must pick a and b such that Magnetostatics Notes | EduRev


Example 10: Find the vector potential of an infinite solenoid with n turns per unit length, radius R, and current I.

Since Magnetostatics Notes | EduRev where φ is the flux of Magnetostatics Notes | EduRev through the loop in question.
Inside solenoid:
Magnetostatics Notes | EduRev
Outside solenoid:
Magnetostatics Notes | EduRev


Multiple Expansion of Vector Potential 

Magnetostatics Notes | EduRev

We can derive approximate formula for the vector potential of a localized current distribution, valid at distant points. We can always write the potential in the form of a power series in 1/r, where r is the distance to the point in question. Thus we can always write

Magnetostatics Notes | EduRev

First term, monopole Magnetostatics Notes | EduRev (no magnetic monopole) 

Second term, dipole
Magnetostatics Notes | EduRev
Magnetostatics Notes | EduRev

where Magnetostatics Notes | EduRev is the magnetic dipole moment:  Magnetostatics Notes | EduRev 
where Magnetostatics Notes | EduRev is area vector
Thus
Magnetostatics Notes | EduRev
Hence
Magnetostatics Notes | EduRev
Note: (a) When a magnetic dipole is placed in a uniform magnetic field Magnetostatics Notes | EduRev 

net force on the dipole is zero and it experiences a torque Magnetostatics Notes | EduRev 

(b) In non-uniform field, dipoles have net force Magnetostatics Notes | EduRev and torque 

Magnetostatics Notes | EduRev 
(c) Energy of an ideal dipole Magnetostatics Notes | EduRev in an magnetic field Magnetostatics Notes | EduRev 

(d) Interaction energy of two dipoles separated by a distance Magnetostatics Notes | EduRev is
Magnetostatics Notes | EduRev


Example 11: A phonograph record of radius R, carrying a uniform surface charge σ is rotating at constant angular velocity ω. Find its magnetic dipole moment.

Magnetic moment of a ring of radius r and thickness dr is, dm = Iπr2 where
I = σvdr= σrωdr
Magnetostatics Notes | EduRev


Example 12: A spherical shell of radius R, carrying a uniform surface charge σ, is set spinning at angular velocity ω. Find its Magnetic dipole moment.

Magnetostatics Notes | EduRev

The total charge on the shaded ring is
dq = σ (2π R sinθ ) Rdθ
Time for one revolution is
Magnetostatics Notes | EduRev
⇒Current in the ring Magnetostatics Notes | EduRev
Area of the ring =π(R sin θ)2 , so the magnetic moment of the  ring is
Magnetostatics Notes | EduRev


Magnetisation Magnetostatics Notes | EduRev

If a piece of magnetic material is examined on an atomic scale we will find tiny currents: electrons orbiting around nuclei and electrons spinning about their axes. For macroscopic purpose, these current loops are so small that we may treat them as magnetic dipoles. Ordinarily they cancel each other out because of the random orientation of the atoms. But when a magnetic field is applied, a net alignment of these magnetic dipoles occurs, and medium becomes magnetically polarized, or magnetized.
Magnetization Magnetostatics Notes | EduRev  is magnetic dipole moment per unit volume.

The Field of a Magnetized Object (Bound Currents) 

Consider a piece of magnetized material with magnetization Magnetostatics Notes | EduRev

Then the vector potential of a single dipole Magnetostatics Notes | EduRev is given by
Magnetostatics Notes | EduRev
Magnetostatics Notes | EduRev

In the magnetized object, each volume element dτ ′ carries a dipole moment Magnetostatics Notes | EduRev so the total vector potential is 

Magnetostatics Notes | EduRev

The equation can be written as
Magnetostatics Notes | EduRev
The first term is like potential of a volume current
Magnetostatics Notes | EduRev
while the second term is like potential of a surface current,
Magnetostatics Notes | EduRev
whereMagnetostatics Notes | EduRev is the normal to the unit vector. With these definitions, the field of a magnetized object is
Magnetostatics Notes | EduRev
This means the potential(and hence also the field) of a magnetized object is the same as would be produced by a volume current Magnetostatics Notes | EduRev throughout the material, plus a surface current Magnetostatics Notes | EduRev on the boundary. We first determine these bound currents, and then find the field they produce. 


Example 13: An infinitely long circular cylinder carries a uniform magnetization Magnetostatics Notes | EduRev

parallel to its axis. Find the magnetic field (due to Magnetostatics Notes | EduRev ) inside and outside the cylinder.

Magnetostatics Notes | EduRev 

The field is that of a surface current Magnetostatics Notes | EduRev that is the case of a solenoid,  

So the field outside is zero.
Field inside is:

Magnetostatics Notes | EduRev 


Example 14: A long circular cylinder of radius R carries a magnetization Magnetostatics Notes | EduRev 

where K is a constant; r is the distance from the axis. Find the magnetic field due to 

Magnetostatics Notes | EduRev for points inside and outside the cylinder.

Magnetostatics Notes | EduRev

So the bound current flows up the cylinder, and returns down the surface.
Magnetostatics Notes | EduRev

Outside point: Ienc = 0⇒B =0
Inside point:
Magnetostatics Notes | EduRev


The Auxiliary field Magnetostatics Notes | EduRev

Ampere’s Law in in presence of Magnetic Materials  In a magnetized material the total current can be written as Magnetostatics Notes | EduRev where Magnetostatics Notes | EduRev is bound current and
Magnetostatics Notes | EduRev is free current. 

Magnetostatics Notes | EduRev

The quantity in parentheses is designated by the letter Magnetostatics Notes | EduRev
Magnetostatics Notes | EduRev
In integral form Magnetostatics Notes | EduRev where Magnetostatics Notes | EduRev is the total free current passing through the amperian loop. 

Magnetostatics Notes | EduRev  plays a role in magnetostatic analogous to Magnetostatics Notes | EduRev  in electrostatic: Just as Magnetostatics Notes | EduRev 

 allowed us to write Gauss's law in terms of the free charge alone, Magnetostatics Notes | EduRev permits us to express Ampere's law in terms of the free current alone- and free current is what we control directly. Note: When we have to find  Magnetostatics Notes | EduRev  or Magnetostatics Notes | EduRev  in a problem involving magnetic materials, first look for symmetry. If the problem exhibits cylindrical, plane, solenoid, or toroidal symmetry, then we can get Magnetostatics Notes | EduRev directly from the equation Magnetostatics Notes | EduRev

Magnetic Susceptibility and Permeability

For most substances magnetization is proportional to the field Magnetostatics Notes | EduRev where χm is magnetic susceptibility of the material.
Magnetostatics Notes | EduRev where(μ = μ0μr= μ0(1 + χm) is permeability   

of material.

Boundary Condition Magnetostatics Notes | EduRev

Magnetostatics Notes | EduRev

The boundary between two medium is a thin sheet of free surface current Kf . The Ampere’s law states that

Magnetostatics Notes | EduRev

Since
Magnetostatics Notes | EduRev

Thus

Magnetostatics Notes | EduRev


Example 15: A current I flows down a long straight wire of radius a. If the wire is made of linear material with susceptibility χm , and the current is distributed uniformly, what is the magnetic field a distance r from the axis? Find all the bound currents. What is the net bound current following down the wire? 

Magnetostatics Notes | EduRev

Magnetostatics Notes | EduRev

Magnetostatics Notes | EduRev

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