5.1.1 Magnetic Fields

Chapter 5. Magnetostatics

5.1 The Lorentz Force Law

5.1.1 Magnetic Fields

Consider the forces between charges in motion

Attraction of parallel currents and Repulsion of antiparallel ones:

 How do you explain this?

 Does charging up the wires make simply the electrical repulsion of like charges? The wires are in fact electrically neutral!

 It is not electrostatic in nature.

 A moving charge generates a Magnetic field B.

If you grab the wire with your right hand-thumb in the direction of the current  your fingers curl around in the direction of the magnetic field.

How can such a field lead to a force of attraction on a nearby parallel current?

How do you calculate the magnetic field?

5.1.2 Magnetic Forces

The magnetic force in a charge Q, moving with velocity v in a magnetic field B, is In the presence of both electric and magnetic fields, the net force on Q would be

 Lorentz force law

 It is a fundamental axiom justified in experiments.

 Magnetic forces, which do not work!

 For if Q moves an amount dl = v dt, the work done is

 Magnetic forces may alter the direction in which a particle moves, but they cannot speed it up or slow it down.

Stationary charges produce electric fields that are constant in time  Electrostatics

Steady currents produce magnetic fields that are constant in time  Magnetostatics

Lorentz Forces:

Example 5.2 Cycloid motion

Let's think it through qualitatively, first.

 Initially, the particle is at rest, so the magnetic force is zero.

 The electric field accelerates the charge in the z-direction.

 As it picks up speed, a magnetic force develops.

 The magnetic field pulls the charge around to the right.

 The faster it goes, the stronger magnetic force becomes; it curves the particle back around towards the y axis.

 The charge is moving against the electrical force, so it begins to slow down.

 the magnetic force then decreases and the electrical force takes over, bringing the charge to rest at point a.

 There the entire process commences anew, carrying the particle over to point b, and so on.

Now let's do it again quantitatively.

(cyclotron frequency)

(General solution)

At t = 0:

At z =0: _{E mag} E E

F F R

B B

 

     

This is the formula for a circle, of radius R, whose center (0, Rt, R)

travels in the y-direction at a constant speed, v = E/B.

 Cycloid motion

5.1.3 Currents 

When a line charge  traveling down at velocity v 

The magnetic force on a segment of current-carrying wire is

For a steady current (constant in magnitude) along the wire 

When charge flows over a surface with surface current density, K:

 current per unit width-perpendicular-to-flow

 (surface charge density)

When charge flows over a volume with volume current density, J:

 current per unit area-perpendicular-to-flow

Continuity Equation of current density

 The current crossing a surface S can be written as



The total charge per unit time leaving a volume V is

Because charge is conserved,

 Continuity equation (Local charge conservation)

For a steady current,

/ t 0



  

0

J 0

0

S j

ds I



    

 (Kirchhoff’s current law)

J 0

  

5.2 The Biot-Savart Law

5.2.1 Steady Currents

Steady current  A continuous flow that has been going on forever,

without change and without charge piling up anywhere.

For a steady current, constant magnetic fields: Magnetostatics 

5.2.2 The Magnetic Field of a Steady Current

 Biot-Savart law

The integration is along the current path in the direction of the flow.

(Permeability of free space) (Unit of B, Tesla)

Biot-Savart law in magnetostatics plays a role analogous to Coulomb's law in electrostatics For surface and volume currents

 Biot-Savart law

The Biot-Savart Law

Example 5.5

points out of the page, and has the magnitude of

In the case of an infinite wire,

As an application, let's find the force of attraction between two wires carrying currents, I₁ and I₂. The field at (2) due to (1) is  It points into the page.

It is directed towards (1)

The force per unit length  It is attractive to each other.

5.3 The Divergence and Curl of B:

The magnetic field of an infinite straight wire looks to have a nonzero curl.

 Let’s calculate the Curl of B.

5.3.1 Straight-Line Currents

 It is independent of s.

If we use cylindrical coordinates (s, , z), with the current flowing along the z axis,

Now suppose we have a bundle of straight wires:

Applying Stokes' theorem 

 But, this derivation is seriously flawed by the restriction to infinite straight line currents.

 Let’s use the Biot-Savart Law for general derivation.

  B  B

( We knew it for the infinite long wire from Biot-Savart law)

The Divergence of B:

The Biot-Savart law for the general case of a volume current is

Applying the divergence with respect to the unprimed coordinates:

   B 0 0

   B

The Curl of B:

Applying the curl with respect to the unprimed coordinates:

B 

J

 

The x component, in particular, is

For steady currents the divergence of J is zero  Note the switching by putting - sign from

We can have the boundary surface large enough, so the current is zero on the surface (all current is safely inside)

= 0

?  Ampere's Law

B 

J

 

5.3.3 Applications of Ampere's Law

B 

J

   B   d l 

I

Which direction through the surface corresponds to a positive current?

 Keep the 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.

Electrostatics : Coulomb law  Gauss’s law

Magnetostatics: Biot-Savart law  Ampere’s law

Total current passing through the surface

Amperian loop

Like Gauss's law, Ampere's law is useful

only when the symmetry of the problem enables you to pull B outside the integral.

Ampere's Law

Example 5.7

B   d l 

I



Find the magnetic field a distance s from a long straight wire.

Example 5.8

Example 5.9

consisting of n closely wound turns per unit length on a cylinder of radius R and carrying a steady current I.

B of an infinite, closely wound solenoid runs parallel to the axis:

B can only have a y-component:

5.3.4 Comparison of Magnetostatics and Electrostatics

The divergence and curl of the electrostatic field are

The divergence and curl of the magnetostatic field are

The force law is

 Typically, electric forces are enormously larger than magnetic ones.

 These 4 equations are Maxwell's Equations for static electromagnetics

5.4 Magnetic Vector Potential

5.1.1 The Vector Potential

In electrostatics,    E 0  Scalar potential (V) 0

    B

In magnetostatics, B   A  Vector potential (A)

E   V

So far, for a steady current where the Magnetic field is defined by the Biot-Savart law:

This Magnetic field given by the Biot-Savart law always satisfies the relations of

(as long as the current is steady, ) (always zero, even for not steady current)

/ 0

J



     

(If the current is not steady,  B ₀J₀(D/t))

B 

J

 

The relations of , called Ampere’s Law, is very useful, just like the Gauss’s Law in E.

The relations of is always satisfied for any current flow, therefore we may put the B field, either

B   A

( )

B   A   

(for any arbitrary vector A)

(for any arbitrary vector A and any scalar , since )

Or,    0

(Note) The name is “potential”, but A cannot be interpreted as potential energy per unit charge.

 

B A A A  J

         

B    A WHY not ? B    ( A    )

B 

J

 

Actually we want to use the Ampere’s Law, , to find out B field.

B    A (1) Let’s use the form of

2

A 

J

  

If we choose A so as to eliminate the divergence of A:

   A 0

B 

J

 

We know how to solve it, just like the electrostatic potential problems.

2

A



  

 In summary, the definition B    A under the condition of    ^{A 0}

make it possible to transform the Ampere’s law into a Poisson’s equation on A and J !

If goes to zero at infinite,

If J goes to zero at infinite,

(If the current does not go to zero at infinite, we have to find other ways to get A.) If I goes to zero at infinite,

If K goes to zero at infinite, Ampere’s Law

B    A WHY not ? B    ( A    )

(2) Now consider the other form,



Suppose that a vector potential A₀satisfying B    A₀, but it is not divergenceless,  A₀  0.

If we add to A₀ the gradient of any scalar , A  A₀  



since )

   0





(  A  



, A is also satisfied B    A.

The divergence of A is

     A A

 



We can accommodate the condition of

as long as a scalar function  can be found that satisfies,

   A 0    

 A

But this is mathematically identical to Poisson's equation, thus

We can always find  to make

   A 0

 Therefore, let’s use the simple form of with the divergenceless vector potential, B    A

   A 0

May we use a scalar potential, , just like used in ?

The introduction of Vector Potential (A) may not be as useful as V,

 It is still a vector with many components.

It would be nice if we could get away with a scalar potential, for instant,

 But, this is incompatible with Ampere's law, since the curl of a gradient is always zero 

B   U B 

J

 

0

B U

   

B   U E   V

 Such a magnetostatic scalar potential could be used,

if you stick scrupulously to simply-connected current-free regions (J = 0).

Suppose that .

Show, by applying Ampere's law to a path that starts at a and circles the wire returning to b

that the scalar potential cannot be single-valued, , even if they are the same point (a = b).

Prob. 5-29

(by the gradient theorem),

5.4.2 Summary; Relations of B – J – A

J

A

From just two experimental observations:

(1) the principle of superposition - a broad general rule

(2) Biot-Savert’s law - the fundamental law of magnetorostatics.

Poisson's equation Ampere’s Law

Stoke’s theorem

B – J – A

2

A



  

  B 

J

B   A

B

E –  – V B – J – A

5.4.2 Magnetostatic Boundary Conditions

(Normal components)

(Tangential components)

For an amperian loop running perpendicular to the current,

For an amperian loop running parallel to the current,

Finally, we can summarize in a single formula:

The vector potential is continuous across any boundary:

(Because, guarantees the normal continuity; the tangential one.) (the flux through an amperian loop of vanishing thickness is zero)

E  B Summary of Boundary conditions:

above below

V  V A

 A

z y x

5.4.3 Multipole Expansion of the Vector Potential

If we want an approximate formula for the vector potential of a localized current distribution, valid at distant points, a multipole expansion is required.

(Eq. 3.94) Multipole expansion means

 to write the potential in the form of a power series in 1/r, if r is sufficiently large.

monopole dipole quadrupole

monopole dipole quadrupole

Just for comparison:

Multipole Expansion of the Vector Potential

monopole dipole quadrupole

(Magnetic Monopole Term)  Always zero!

 For the integral is just the total displacement around a closed path,

 This reflects the fact that there are (apparently) no magnetic monopoles in nature.

(Magnetic Dipole Term)  It is dominant !

 Magnetic dipole moment

Magnetic Dipole field

Magnetic field of a (pure) dipole moment placed at the origin.

Field B of a "pure" dipole

Field B of a “physical" dipole

Summary