Sources of the Magnetic Field Magnetic Induction Fall

Sources of the Magnetic Field & Magnetic Induction Fall 2006

Remember the wire?

Try to remember…

The “Coulomb’s Law” of Magnetism A Vector Equation … duck

For the Magnetic Field, current “elements” create the field. This is the Law of Biot-Savart

Magnetic Field of a Straight Wire We intimated via magnets that the Magnetic field associated with a straight wire seemed to vary with 1/d. ¢ We can now PROVE this! ¢

From the Past Using Magnets

Right-hand rule: Grasp the element in your right hand with your extended thumb pointing in the direction of the current. Your fingers will then naturally curl around in the direction of the magnetic field lines due to that element.

Let’s Calculate the FIELD Note: For ALL current elements in the wire: ds X r is into the page

The Details

Moving right along 1/d

A bit more complicated A finite wire

P 1 r p-q q ds

More P 1

P 2

APPLICATION: Find the magnetic field B at point P

Center of a Circular Arc of a Wire carrying current

More arc… ds

The overall field from a circular current loop Sorta looks like a magnet!

Iron

Howya Do Dat? ? No Field at C

Force Between Two Current Carrying Straight Parallel Conductors Wire “a” creates a field at wire “b” Current in wire “b” sees a force because it is moving in the magnetic field of “a”.

The Calculation

Definition of the Ampere The force acting between currents in parallel wires is the basis for the definition of the ampere, which is one of the seven SI base units. The definition, adopted in 1946, is this: The ampere is that constant current which, if maintained in two straight, parallel conductors of infinite length, of negligible circular cross section, and placed 1 m apart in vacuum, would produce on each of these conductors a force of magnitude 2 x 10 -7 newton per meter of length.

Ampere’s Law The return of Gauss

Remember GAUSS’S LAW? ? Surface Integral

Gauss’s Law Made calculations easier than integration over a charge distribution. ¢ Applied to situations of HIGH SYMMETRY. ¢ Gaussian SURFACE had to be defined which was consistent with the geometry. ¢ ¢ AMPERE’S Law is the Gauss’ Law of Magnetism! (Sorry)

The next few slides have been lifted from Seb Oliver on the internet Whoever he is!

Biot-Savart ¢ The “Coulombs Law of Magnetism”

Invisible Summary ¢ Biot-Savart Law (Field produced by wires) l Centre of a wire loop radius R l Centre of a tight Wire Coil with N turns l Distance a from long straight wire l Force between two wires ¢ Definition of Ampere ¢

Magnetic Field from a long wire Using Biot-Savart Law r I B ds Take a short vector on a circle, ds Thus the dot product of B & the short vector ds is:

Sum B. ds around a circular path r I B Sum this around the whole ring ds Circumference of circle

Consider a different path Field goes as 1/r ¢ Path goes as r. ¢ Integral independent of r ¢ i

SO, AMPERE’S LAW by SUPERPOSITION: We will do a LINE INTEGRATION Around a closed path or LOOP.

Ampere’s Law USE THE RIGHT HAND RULE IN THESE CALCULATIONS

The Right Hand Rule. . AGAIN

Another Right Hand Rule

COMPARE Line Integral Surface Integral

Simple Example

Field Around a Long Straight Wire

Field INSIDE a Wire Carrying UNIFORM Current

The Calculation

B R r

Procedure ¢ ¢ Apply Ampere’s law only to highly symmetrical situations. Superposition works. l ¢ ¢ Two wires can be treated separately and the results added (VECTORIALLY!) The individual parts of the calculation can be handled (usually) without the use of vector calculations because of the symmetry. THIS IS SORT OF LIKE GAUSS’s LAW WITH AN ATTITUDE!

The figure below shows a cross section of an infinite conducting sheet carrying a current per unit x-length of l; the current emerges perpendicularly out of the page. (a) Use the Biot–Savart law and symmetry to show that for all points P above the sheet, and all points P´ below it, the magnetic field B is parallel to the sheet and directed as shown. (b) Use Ampere's law to find B at all points P and P´.

FIRST PART Vertical Components Cancel

Apply Ampere to Circuit L B Infinite Extent B

The “Math” 0 B B s= d Infinite Extent B

A Physical Solenoid

Inside the Solenoid For an “INFINITE” (long) solenoid the previous problem and SUPERPOSITION suggests that the field OUTSIDE this solenoid is ZERO!

More on Long Solenoid Field is ZERO! Field looks UNIFORM Field is ZERO

The real thing…. . Finite Length Weak Field Stronger - Leakage ly ir Fa rm fo ni U d el fi

Another Way

Application Creation of Uniform Magnetic Field Region ¢ Minimal field outside ¢ l except at the ends!

Two Coils

“Real” Helmholtz Coils Used for experiments. Can be aligned to cancel out the Earth’s magnetic field for critical measurements.

The Toroid Slightly less dense than inner portion

The Toroid

Induction

Magnetic Flux For a CLOSED Surface we might expect this to be equal to some constant times the enclosed poles … but there ain’t no such thing!

Examples S N

Consider the poor little capacitor… i i CHARGING OR DISCHARGING …. HOW CAN CURRENT FLOW THROUGH THE GAP? ?

Through Which Surface Do we measure the current for Ampere’s Law? I=0

In the gap… DISPLACEMENT CURRENT

From The Demo. .

Faraday’s Experiments ? ?

Insert Magnet into Coil

Remove Coil from Field Region

That’s Strange …. . These two coils are perpendicular to each other

Definition of TOTAL ELECTRIC FLUX through a surface:

Magnetic Flux: F THINK OF MAGNETIC FLUX as the “AMOUNT of Magnetism” passing through a surface. Don’t quote me on this!!!

Consider a Loop ¢ xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx ¢ ¢ Magnetic field passing through the loop is CHANGING. FLUX is changing. There is an emf developed around the loop. A current develops (as we saw in demo) Work has to be done to move a charge completely around the loop.

Faraday’s Law (Michael Faraday) xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx ¢ ¢ ¢ For a current to flow around the circuit, there must be an emf. (An emf is a voltage) The voltage is found to increase as the rate of change of flux increases.

Faraday’s Law (Michael Faraday) xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx We will get to the minus sign in a short time.

Faraday’s Law (The Minus Sign) xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx Using the right hand rule, we would expect the direction of the current to be in the direction of the arrow shown.

Faraday’s Law (More on the Minus Sign) xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx The minus sign means that the current goes the other way. This current will produce a magnetic field that would be coming OUT of the page. The Induced Current therefore creates a magnetic field that OPPOSES the attempt to INCREASE the magnetic field! This is referred to as Lenz’s Law.

How much work? xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx f m e A magnetic field an electric field are intimately connected. )

The Strange World of Dr. Lentz

MAGNETIC FLUX This is an integral over an OPEN Surface. ¢ Magnetic Flux is a Scalar ¢ ¢ The UNIT of FLUX is the l 1 weber = 1 T-m 2 weber

We finally stated FARADAY’s LAW

From the equation Lentz

Flux Can Change ¢ ¢ If B changes If the AREA of the loop changes Changes cause emf s and currents and consequently there are connections between E and B fields These are expressed in Maxwells Equations

Maxwell’s Equations (Next Course. . Just a Preview!) Gauss Faraday

Another View Of That damned minus sign again …. . SUPPOSE that B begins to INCREASE its MAGNITUDE INTO THE PAGE xxxxxxxxxxxxxxx xxxxxxxxxxxxxxx ¢ ¢ ¢ The Flux into the page begins to increase. An emf is induced around a loop A current will flow That current will create a new magnetic field. THAT new field will change the magnetic flux.

Lenz’s Law Induced Magnetic Fields always FIGHT to stop what you are trying to do! i. e. . . Murphy’s Law for Magnets

Example of Nasty Lenz The induced magnetic field opposes the field that does the inducing!

Don’t Hurt Yourself! The current i induced in the loop has the direction such that the current’s magnetic field Bi opposes the change in the magnetic field B inducing the current.

Let’s do the Lentz Warp again !

Lenz’s Law An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux that induces the current. (The result of the negative sign!) … OR The toast will always fall buttered side down!

An Example The field in the diagram creates a flux given by FB=6 t 2+7 t in milli. Webers and t is in seconds. (a) What is the emf when (b) t=2 seconds? (c) (b) What is the direction (d) of the current in the (e) resistor R?

This is an easy one … Direction? B is out of the screen and increasing. Current will produce a field INTO the paper (LENZ). Therefore current goes clockwise and R to left in the resistor.

Figure 31 -36 shows two parallel loops of wire having a common axis. The smaller loop (radius r) is above the larger loop (radius R) by a distance x >> R. Consequently, the magnetic field due to the current i in the larger loop is nearly constant throughout the smaller loop. Suppose that x is increasing at the constant rate of dx/dt = v. (a) Determine the magnetic flux through the area bounded by the smaller loop as a function of x. (Hint: See Eq. 30 -29. ) In the smaller loop, find (b) the induced emf and (c) the direction of the induced current. v

B is assumed to be constant through the center of the small loop and caused by the large one. q

The calculation of Bz q

More Work In the small loop: dx/dt=v

Which Way is Current in small loop expected to flow? ? B q

What Happens Here? ¢ ¢ ¢ Begin to move handle as shown. Flux through the loop decreases. Current is induced which opposed this decrease – current tries to reestablish the B field.

moving the bar

Moving the Bar takes work v

What about a SOLID loop? ? Energy is LOST BRAKING SYSTEM METAL Pull

Back to Circuits for a bit ….

Definition Current in loop produces a magnetic field in the coil and consequently a magnetic flux. If we attempt to change the current, an emf will be induced in the loops which will tend to oppose the change in current. This this acts like a “resistor” for changes in current!

Remember Faraday’s Law Lentz

Look at the following circuit: Switch is open ¢ NO current flows in the circuit. ¢ All is at peace! ¢

Close the circuit… After the circuit has been close for a long time, the current settles down. ¢ Since the current is constant, the flux through the coil is constant and there is no Emf. ¢ ¢ Current is simply E/R (Ohm’s Law)

Close the circuit… ¢ ¢ When switch is first closed, current begins to flow rapidly. The flux through the inductor changes rapidly. An emf is created in the coil that opposes the increase in current. The net potential difference across the resistor is the battery emf opposed by the emf of the coil.

Close the circuit…

Moving right along …

Definition of Inductance L UNIT of Inductance = 1 henry = 1 T- m 2/A FB is the flux near the center of one of the coils making the inductor

Consider a Solenoid l n turns per unit length

So…. Depends only on geometry just like C and is independent of current.

Inductive Circuit i ¢ ¢ ¢ Switch to “a”. Inductor seems like a short so current rises quickly. Field increases in L and reverse emf is generated. Eventually, i maxes out and back emf ceases. Steady State Current after this.

THE BIG INDUCTION ¢ ¢ As we begin to increase the current in the coil The current in the first coil produces a magnetic field in the second coil Which tries to create a current which will reduce the field it is experiences And so resists the increase in current.

Back to the real world… Switch to “a” i

Solution

Switch position “b”

VR=i. R ~current Max Current Rate of increase = max emf

. n o ti he t e v l So p o o l ua q e

IMPORTANT QUESTION Switch closes. ¢ No emf ¢ Current flows for a while ¢ It flows through R ¢ Energy is conserved (i 2 R) ¢ WHERE DOES THE ENERGY COME FROM? ?

For an answer Return to the Big C E=e 0 A/d ¢ ¢ +dq +q ¢ -q ¢ We move a charge dq from the (-) plate to the (+) one. The (-) plate becomes more (-) The (+) plate becomes more (+). d. W=Fd=dq x E x d

The calc The energy is in the FIELD !!!

What about POWER? ? power to circuit Must be d. WL/dt power dissipated by resistor

So Energy stored in the Capacitor

WHERE is the energy? ? l

Remember the Inductor? ? ? ? ? ? ?

ENERGY IN THE FIELD TOO!

IMPORTANT CONCLUSION A region of space that contains either a magnetic or an electric field contains electromagnetic energy. ¢ The energy density of either is proportional to the square of the field strength. ¢