Advanced EM - Master in Physics 2011-2012 1 Magnetic potential and field of a SOLENOID Infinite length N spires/cm Current I Radius R The problem -for

Advanced EM - Master in Physics 2011-2012

1

Magnetic potential and field of a SOLENOID

• Infinite length

•N spires/cm

•Current I

•Radius R

)0,,)0,cos,sin( ( yx AAJJ AJ

The problem -for Ax –is mathematically similar to finding the electrostatic potential Φ with a surface charge ρ0sinθ.

This type of surface charge is what is given by two parallel cylinders of uniform and equal, but of opposite sign, charge densities, with their axes displaced by an infinitesimal distance d: the calculation is shown below. Let V be the potential of a cylinder of radius R and charge density uniform. Two such cylinders of the same radius and charge density opposite in sign have a superficial charge density : sin00 ddr

Rr

y

y

r

yr

yr

y

rd

y

V

rfor )/1(ln(

Rrfor 22

2

2

A

Their potential has a

dy

V

behavio

ur

J


2

Owing to the azimutal symmetry of the system, we can calculate the Φ for x=0, where y=r . We get a r dependence of Φ ~r inside the wire and ~(1/r) outside.

Then, A being always along the direction of the θ axis, it turns around the solenoid axis like the current does. It varies with r like r for r<R and like 1/r for r>R. Once we put all the coefficients right, we obtain:

Rrfor 2

Rrfor 2

2

cr

NIR

c

NIr

A

A

Having A written as a function of r we can calculate its curl, i.e.

B.

Rrfor 0

Rrfor 4

,0,0(

)1

()()1

(

{)

z

z

z

rzrz

Bc

NIB

B

A

rr

A

r

A

r

A

z

A

z

AA

r

AB

krAB

Having Bz finite inside the solenoid and at its radius

suddenly zero is not realistic. But… also an infinite solenoid is not quite realistic. And anyway the coil has a finite thickness.

Note at this point an important physical fact: outside the solenoid the vector potential is not zero nor constant, BUT:

θ


3

BUT:

r

constA

While not being trivial still has curl equal zero. On the other hand it is quickly verified that this A has zero divergence. We are then led by this paradoxical result to ask ourselves where

the Physics is, in the field B or in the potential A. The answer has been given by the Bohm – Aharonov experiment, 1956. Which has already been described.


4

EM fields varying in time

This is the term added by Maxwell to preserve charge conservation. It allows the equations to become “Wave equations”.

With this new term the electrical and magnetic fields are now coupled through their dependence on time. Without it, a variation of B with time would have caused a variation of E. End of the story. Now, a variation of E changes B as well: the process is self-sustaining.

Now a question comes out immediately: beforehand (in electrostatics) we knew that we could express E as the gradient of a potential and B as the curl of a “vector potential”. How does that change now?

AB

B

0 This still holds, then

this is still validWe then insert it in the second Maxwell equation and obtain

)( 1)(1

tcc

A

t

AE

}{14

0

1

4

tcc

tc

ΕJΒ

Β

ΒΕ

Ε


5

This equation is satisfied, by the formula on the right: the 3rd and 2nd EofM are satisfied since the condition on the curl of E is satisfied, and this determines E

tc

AE

1

up to the gradient of any function! We call that function Φ; it will be determined by the 1st EofM which, upon substitution of this value for E becomes

)( 1

tc

AE

JAA

A

AJAA

AJA

BE

A

2

ctctc

tctcc

tctcc

tc

4)

1(

1

,reordering and,

)(114)(

)(114)(

EofM4th in the and gsubstituin while,

4)(1

2

2

2

2

2

2

2

2

2

2

2

We still have to choose the gauge. For the time-dependent case we choose the Lorentz gauge, defined by:

gauge. Lorentz The 1

tc

A

Vector formula


6

Inserting now the Lorentz gauge condition in the last 2 equations in Φ and in A they become much simpler:

JA

Actc

tc

41

41

2

2

22

2

2

22

Which is (are, actually) the classical wave equations. With wave velocity = c !!!

These equations are of course NOT independent: beside the Lorentz gauge which has been used to obtain them, and

constrains Φ and A, J and ρ are also related by the charge conservation. In formulas:

on conservati Charge t

gauge Lorentz The 1

0J

A

tc


7

Faraday’s law

The second EofM, also called Faraday’s induction law, usually given in its differential form:

tc

B

E1

can also be written in integral form (using the Stokes theorem in the passage from line to surface integral) if we define the “electromotive force” E

)(11

1)(

)( BnB

nB

nElE

S

S

SS

tcdS

tc

dStc

dSd

E

Now, let the closed line over which we calculate the line integral be a real electric wire. The electromotive force E is

generated by the variation (in time) of the flux of B through any surface enclosed by the closed line. Such variation can be generated in three different ways:

•Varying the current through the magnets.

•Moving the circuit.

•Moving the magnets.

Matter of fact, what is actually measurable is the force on a charge in the wire, so the equation becomes:

lBv

ElF dc

dq

)(1E


8

In the laboratory reference system, if the circuit is moving with

velocity v, a free charge inside it will feel the force )( BvEF q

1. What changes in time is the current running in the magnets. Then:

tctcdS

tc

dSd

S

SS

S

1

)(11

)(

)( nBnB

nElEE

Case verified

2. The circuit moves with velocity v while the magnetic field does not change.

V

0

t

B0v

BlvlBv

)()( ddc

E

Then a free charge (electron) will feel the Lorentz force:

( V х dl ) is a vector which is

normal to the surface element dS (yellow in the drawing) covered by the

circuit in the time dt.

Then: E = -(1/c) · [time variation of

the flux of B due to the change of surface covered by the circuit (in blue and in yellow in the drawing)].

Let us now study these 3 different cases and see what we get from them.


9

3. The magnets are moved: the currents which generate B do not change, the circuit sits still, but the magnets are moved. This case is a sort of bridge between the two previous ones: from the

mathematical point of view in the formulas it is B that has changed; but it is also identical to the second case, because if what has moved is the circuit or the magnets the result is the same.

BUT…

But, the choice of the reference system, causes the choice in the mathematics of either of the term

BvA

c

1E

c

force Lorentz theofor t

in the calculation of E. The result is, however, the same!

Another remark on the case a): currents varying with time. Let us imagine a transformer.

An electromotive force is generated on the circuit C when the current is changed on A. But, where C is located there is hardly a magnetic field. There is, however a significant vector potential.

CA

r

1- ~ A


10

The conservation of energy

We have already seen the mathematical expression for the charge conservation

0J

t

Since we know that energy is also conserved, we would like to find another similar equation for the energy conservation – of the electromagnetic field, of course. We are therefore looking for two physical quantities, let us call

them U and S, which can be related to energy density (of the EM field) and to energy flow, which satisfy the following equation.

charges}on donework {

St

U

The term on the right-hand side of this equation has the form:

JE charges}on donework {

What we shall do now is to try and find a formula – in which only the fields appear - which looks like the first term in the equation above which states the conservation, i.e. an equation which has the form:

“(the derivative of a scalar plus the divergence of a vector) = -

E·J”.

We shall do that starting from E·J, by substituting J with its expression taken from the 4th EofM:

)(4

1

tc

E

BJ


11

Here goes the calculation: we shall use the vector formula

)(8

1)

4(

2

1)()(

4

1

)(4

1

22

2

][

][

BEt

ct

Ecc

tc

BE

EBEB

EEBEJE

FormulaVector )()()( KRRKKR

The calculation goes as follows:

This is Poynting’s theorem. Note that it equals E·J to the sum of a time derivative of a scalar + the divergence of a vector, precisely what we were looking for. It is therefore

immediate to interpret this formula as follows:

flowenergy : vectorPoynting )(4

densityEnergy )(8

1 22

BES

c

BEU

And the equation for the conservation of energy is:

0JES

t

U


12

So, we have now, in regime of full time dependence of the fields and of the charges and currents, found a formula for the energy distribution of the electromagnetic field. But… hadn’t we already met a formula for this energy density? Well, we had found two. We write for simplicity only the electric density. The general formula adds the electric and the magnetic energies

Vd

dVEU

V

space

)()(2

1U

and 8

1 2

rr

Only one of them is valid in general: the other one (that

with the charge density ρ ) is only valid in the electrostatic case.

Examples of Poynting’s vector

.

E

x

.

.j

k

B

S

The energy enters the wire from outside!!!


13

The charging of a capacitor

What we want to explain in this slide is how are the fields and the potentials in the case of a parallel plate capacitor being charged with

a constant current. NB: between the capacitor plates, J=0 !!

J

h

R

B

ES

E and dE/dt are directed along the vertical axis. To

find out how is B directed in the gap, let’s write

dt

dE

c

rB

dt

dEr

crB

dStc

dSdS S

2

12

1)(

2

nE

nBlB

A is the integral of J, therefore

both Ax and Ay are zero. Az

only depends on r, and so the

curl of A is directed along the θ axis. Then S, the Poynting’s vector, is directed

along the axis orthogonal to E

and B, therefore is directed towards the axis of the capacitor, and has value

dt

dEE

rBE

cS

84

The electrostatic energy has uniform density between the plates, U=(1/8π)E2, the total energy is R2πhU and its rate of change is (Rh/4)E(dE/dt) . The S vector is directed radially from outside towards the capacitor gap: ALL THE ENERGY STORED IN THE CAPACITOR HAS ENTERED LATERALLY FROM THE OUTSIDE AIR!

Documents

Advanced EM - Master in Physics 2011-2012 1 Magnetic potential and field of a SOLENOID Infinite length N spires/cm Current I Radius R The problem -for