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Magnetostatics Notes | Study Electricity & Magnetism - Physics

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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 | Study Electricity & Magnetism - Physicsis, Magnetostatics Notes | Study Electricity & Magnetism - PhysicsThis is known as Lorentz force law. 

In the presence of both electric and magnetic fields, the net force on Q would be:  
Magnetostatics Notes | Study Electricity & Magnetism - Physics

Current in a Wire

Magnetostatics Notes | Study Electricity & Magnetism - Physics

A line charge λ traveling down a wire at a speed Magnetostatics Notes | Study Electricity & Magnetism - Physics constitutes a current Magnetostatics Notes | Study Electricity & Magnetism - Physics 

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

Magnetostatics Notes | Study Electricity & Magnetism - Physics

since Magnetostatics Notes | Study Electricity & Magnetism - Physics  points in the same direction
Magnetostatics Notes | Study Electricity & Magnetism - Physics
Surface Current Density

Magnetostatics Notes | Study Electricity & Magnetism - Physics

When charge flows over a surface, we describe it by the surface current Magnetostatics Notes | Study Electricity & Magnetism - Physics

Magnetostatics Notes | Study Electricity & Magnetism - Physics is the current per unit width-perpendicular to flow.
Also Magnetostatics Notes | Study Electricity & Magnetism - Physics where is σ surface charge density and Magnetostatics Notes | Study Electricity & Magnetism - Physics is its velocity.

Magnetic force on surface current Magnetostatics Notes | Study Electricity & Magnetism - Physics

Volume Current Density

Magnetostatics Notes | Study Electricity & Magnetism - Physics

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

Magnetostatics Notes | Study Electricity & Magnetism - Physics is the current per unit area-perpendicular to flow. 

Also Magnetostatics Notes | Study Electricity & Magnetism - Physics where ρ is volume charge density and Magnetostatics Notes | Study Electricity & Magnetism - Physics is its velocity.

Magnetic force on volume current
Magnetostatics Notes | Study Electricity & Magnetism - Physics
Current crossing a surface S is Magnetostatics Notes | Study Electricity & Magnetism - Physics
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 | Study Electricity & Magnetism - Physics

FE and BA are parallel to magnetic induction Magnetostatics Notes | Study Electricity & Magnetism - Physics. Magnetic force on each of  them will be zero. DE and CB are perpendicular to Magnetostatics Notes | Study Electricity & Magnetism - Physics. 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 | Study Electricity & Magnetism - Physics


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 | Study Electricity & Magnetism - PhysicsConsider an element of length dl of the wire. The Magnetostatics Notes | Study Electricity & Magnetism - Physics force on this element is obtained by
Magnetostatics Notes | Study Electricity & Magnetism - Physics
 (Horizontal component cancels only perpendicular component add up).


Continuity Equation 

The total charge per unit time leaving a volume V is Magnetostatics Notes | Study Electricity & Magnetism - Physics
Because charge is conserved, whatever flows out through the surface must come at the expense of that remaining inside:
Magnetostatics Notes | Study Electricity & Magnetism - Physics
(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 | Study Electricity & Magnetism - Physics

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 | Study Electricity & Magnetism - Physics and hence the continuity equation becomes:
Magnetostatics Notes | Study Electricity & Magnetism - Physics

Biot-Savart Law

Magnetostatics Notes | Study Electricity & Magnetism - Physics

The magnetic field of a steady line current is given by
Magnetostatics Notes | Study Electricity & Magnetism - Physics

where
Magnetostatics Notes | Study Electricity & Magnetism - Physics
For surface and volume current Biot-Savart law becomes:  
Magnetostatics Notes | Study Electricity & Magnetism - Physics
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 | Study Electricity & Magnetism - Physics

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

  1. Magnetic field a distance r from a long straight wire carrying a steady current I is
    Magnetostatics Notes | Study Electricity & Magnetism - Physics 
  2. Magnetic field a distance r from a infinite wire carrying a steady current I is:
    Magnetostatics Notes | Study Electricity & Magnetism - Physics
  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 | Study Electricity & Magnetism - Physics
  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 | Study Electricity & Magnetism - Physics 

Magnetic field due to Toroid is
Magnetostatics Notes | Study Electricity & Magnetism - Physics

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 | Study Electricity & Magnetism - Physics

The field at (2) due to (1) is Magnetostatics Notes | Study Electricity & Magnetism - Physics

Force on (2) is
Magnetostatics Notes | Study Electricity & Magnetism - Physics
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 | Study Electricity & Magnetism - Physics

The field Magnetostatics Notes | Study Electricity & Magnetism - Physics attributable to the segment Magnetostatics Notes | Study Electricity & Magnetism - Physics as shown. As we integrate Magnetostatics Notes | Study Electricity & Magnetism - Physics 

around the loop, Magnetostatics Notes | Study Electricity & Magnetism - Physics sweeps out a cone. The horizontal components cancel, and the vertical components combine to give.

Magnetostatics Notes | Study Electricity & Magnetism - Physics

Magnetostatics Notes | Study Electricity & Magnetism - Physics


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 | Study Electricity & Magnetism - Physics

The force on the two sides cancels.
At the bottom,
Magnetostatics Notes | Study Electricity & Magnetism - Physics 
At the top,
Magnetostatics Notes | Study Electricity & Magnetism - Physics
Thus
Magnetostatics Notes | Study Electricity & Magnetism - Physics


Ampere's Law
Magnetostatics Notes | Study Electricity & Magnetism - Physics

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 | Study Electricity & Magnetism - Physics around a circular path of radius r, centered at the wire, is
Magnetostatics Notes | Study Electricity & Magnetism - Physics

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 | Study Electricity & Magnetism - Physics
In general we can write
Magnetostatics Notes | Study Electricity & Magnetism - Physics
where Ienc is the total current enclosed by the amperian loop.
since
Magnetostatics Notes | Study Electricity & Magnetism - Physics 

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 | Study Electricity & Magnetism - Physics

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 | Study Electricity & Magnetism - Physics 

(1)
Magnetostatics Notes | Study Electricity & Magnetism - Physics
(2)
Magnetostatics Notes | Study Electricity & Magnetism - Physics
Magnetostatics Notes | Study Electricity & Magnetism - Physics


Example 7: Find the magnetic field of an infinite uniform surface current Magnetostatics Notes | Study Electricity & Magnetism - Physics , flowing over the x–y plane.

Magnetostatics Notes | Study Electricity & Magnetism - Physics


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

Magnetostatics Notes | Study Electricity & Magnetism - Physics has only y-component: 

For z >0 , Magnetostatics Notes | Study Electricity & Magnetism - Physics points left (Magnetostatics Notes | Study Electricity & Magnetism - Physics) and for z <0 , Magnetostatics Notes | Study Electricity & Magnetism - Physics  point’s right (Magnetostatics Notes | Study Electricity & Magnetism - Physics ).

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 | Study Electricity & Magnetism - Physics {One Bl from top segment, and the other from bottom}
Magnetostatics Notes | Study Electricity & Magnetism - Physics

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 | Study Electricity & Magnetism - Physics
Since
Magnetostatics Notes | Study Electricity & Magnetism - Physics
For magnetostatic fields,
Magnetostatics Notes | Study Electricity & Magnetism - Physics
if Magnetostatics Notes | Study Electricity & Magnetism - Physics goes to zero at infinity, Magnetostatics Notes | Study Electricity & Magnetism - Physics for volume current.
For line and surface currents,
Magnetostatics Notes | Study Electricity & Magnetism - Physics


Example 8: What current density would produce the vector potential Magnetostatics Notes | Study Electricity & Magnetism - Physics(where K is   a constant), in cylindrical coordinates ?

Magnetostatics Notes | Study Electricity & Magnetism - Physics 
Magnetostatics Notes | Study Electricity & Magnetism - Physics


Magnetostatic Boundary Condition (Boundary is sheet of current, Magnetostatics Notes | Study Electricity & Magnetism - Physics) 

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 | Study Electricity & Magnetism - Physics

Since
Magnetostatics Notes | Study Electricity & Magnetism - Physics
For tangential components
Magnetostatics Notes | Study Electricity & Magnetism - Physics
Thus the component of Magnetostatics Notes | Study Electricity & Magnetism - Physics 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 | Study Electricity & Magnetism - Physics
where Magnetostatics Notes | Study Electricity & Magnetism - Physics 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 | Study Electricity & Magnetism - Physics
For Magnetostatics Notes | Study Electricity & Magnetism - Physics guarantees that the normal component is continuous, and 

Magnetostatics Notes | Study Electricity & Magnetism - Physics in the form
Magnetostatics Notes | Study Electricity & Magnetism - Physics
But the derivative of Magnetostatics Notes | Study Electricity & Magnetism - Physics inherits the discontinuity of Magnetostatics Notes | Study Electricity & Magnetism - Physics 
Magnetostatics Notes | Study Electricity & Magnetism - Physics


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

which means that we must pick a and b such that Magnetostatics Notes | Study Electricity & Magnetism - Physics


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

Since Magnetostatics Notes | Study Electricity & Magnetism - Physics where φ is the flux of Magnetostatics Notes | Study Electricity & Magnetism - Physics through the loop in question.
Inside solenoid:
Magnetostatics Notes | Study Electricity & Magnetism - Physics
Outside solenoid:
Magnetostatics Notes | Study Electricity & Magnetism - Physics


Multiple Expansion of Vector Potential 

Magnetostatics Notes | Study Electricity & Magnetism - Physics

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 | Study Electricity & Magnetism - Physics

First term, monopole Magnetostatics Notes | Study Electricity & Magnetism - Physics (no magnetic monopole) 

Second term, dipole
Magnetostatics Notes | Study Electricity & Magnetism - Physics
Magnetostatics Notes | Study Electricity & Magnetism - Physics

where Magnetostatics Notes | Study Electricity & Magnetism - Physics is the magnetic dipole moment:  Magnetostatics Notes | Study Electricity & Magnetism - Physics 
where Magnetostatics Notes | Study Electricity & Magnetism - Physics is area vector
Thus
Magnetostatics Notes | Study Electricity & Magnetism - Physics
Hence
Magnetostatics Notes | Study Electricity & Magnetism - Physics
Note: (a) When a magnetic dipole is placed in a uniform magnetic field Magnetostatics Notes | Study Electricity & Magnetism - Physics 

net force on the dipole is zero and it experiences a torque Magnetostatics Notes | Study Electricity & Magnetism - Physics 

(b) In non-uniform field, dipoles have net force Magnetostatics Notes | Study Electricity & Magnetism - Physics and torque 

Magnetostatics Notes | Study Electricity & Magnetism - Physics 
(c) Energy of an ideal dipole Magnetostatics Notes | Study Electricity & Magnetism - Physics in an magnetic field Magnetostatics Notes | Study Electricity & Magnetism - Physics 

(d) Interaction energy of two dipoles separated by a distance Magnetostatics Notes | Study Electricity & Magnetism - Physics is
Magnetostatics Notes | Study Electricity & Magnetism - Physics


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 | Study Electricity & Magnetism - Physics


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 | Study Electricity & Magnetism - Physics

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


Magnetisation Magnetostatics Notes | Study Electricity & Magnetism - Physics

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 | Study Electricity & Magnetism - Physics  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 | Study Electricity & Magnetism - Physics

Then the vector potential of a single dipole Magnetostatics Notes | Study Electricity & Magnetism - Physics is given by
Magnetostatics Notes | Study Electricity & Magnetism - Physics
Magnetostatics Notes | Study Electricity & Magnetism - Physics

In the magnetized object, each volume element dτ ′ carries a dipole moment Magnetostatics Notes | Study Electricity & Magnetism - Physics so the total vector potential is 

Magnetostatics Notes | Study Electricity & Magnetism - Physics

The equation can be written as
Magnetostatics Notes | Study Electricity & Magnetism - Physics
The first term is like potential of a volume current
Magnetostatics Notes | Study Electricity & Magnetism - Physics
while the second term is like potential of a surface current,
Magnetostatics Notes | Study Electricity & Magnetism - Physics
whereMagnetostatics Notes | Study Electricity & Magnetism - Physics is the normal to the unit vector. With these definitions, the field of a magnetized object is
Magnetostatics Notes | Study Electricity & Magnetism - Physics
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 | Study Electricity & Magnetism - Physics throughout the material, plus a surface current Magnetostatics Notes | Study Electricity & Magnetism - Physics 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 | Study Electricity & Magnetism - Physics

parallel to its axis. Find the magnetic field (due to Magnetostatics Notes | Study Electricity & Magnetism - Physics ) inside and outside the cylinder.

Magnetostatics Notes | Study Electricity & Magnetism - Physics 

The field is that of a surface current Magnetostatics Notes | Study Electricity & Magnetism - Physics that is the case of a solenoid,  

So the field outside is zero.
Field inside is:

Magnetostatics Notes | Study Electricity & Magnetism - Physics 


Example 14: A long circular cylinder of radius R carries a magnetization Magnetostatics Notes | Study Electricity & Magnetism - Physics 

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

Magnetostatics Notes | Study Electricity & Magnetism - Physics for points inside and outside the cylinder.

Magnetostatics Notes | Study Electricity & Magnetism - Physics

So the bound current flows up the cylinder, and returns down the surface.
Magnetostatics Notes | Study Electricity & Magnetism - Physics

Outside point: Ienc = 0⇒B =0
Inside point:
Magnetostatics Notes | Study Electricity & Magnetism - Physics


The Auxiliary field Magnetostatics Notes | Study Electricity & Magnetism - Physics

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

Magnetostatics Notes | Study Electricity & Magnetism - Physics

The quantity in parentheses is designated by the letter Magnetostatics Notes | Study Electricity & Magnetism - Physics
Magnetostatics Notes | Study Electricity & Magnetism - Physics
In integral form Magnetostatics Notes | Study Electricity & Magnetism - Physics where Magnetostatics Notes | Study Electricity & Magnetism - Physics is the total free current passing through the amperian loop. 

Magnetostatics Notes | Study Electricity & Magnetism - Physics  plays a role in magnetostatic analogous to Magnetostatics Notes | Study Electricity & Magnetism - Physics  in electrostatic: Just as Magnetostatics Notes | Study Electricity & Magnetism - Physics 

 allowed us to write Gauss's law in terms of the free charge alone, Magnetostatics Notes | Study Electricity & Magnetism - Physics 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 | Study Electricity & Magnetism - Physics  or Magnetostatics Notes | Study Electricity & Magnetism - Physics  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 | Study Electricity & Magnetism - Physics directly from the equation Magnetostatics Notes | Study Electricity & Magnetism - Physics

Magnetic Susceptibility and Permeability

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

of material.

Boundary Condition Magnetostatics Notes | Study Electricity & Magnetism - Physics

Magnetostatics Notes | Study Electricity & Magnetism - Physics

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

Magnetostatics Notes | Study Electricity & Magnetism - Physics

Since
Magnetostatics Notes | Study Electricity & Magnetism - Physics

Thus

Magnetostatics Notes | Study Electricity & Magnetism - Physics


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 | Study Electricity & Magnetism - Physics

Magnetostatics Notes | Study Electricity & Magnetism - Physics

Magnetostatics Notes | Study Electricity & Magnetism - Physics

The document Magnetostatics Notes | Study Electricity & Magnetism - Physics is a part of the Physics Course Electricity & Magnetism.
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