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Ch 7 Magnetic Field and Magnetic Forces

PC1432Peter HoDepartment of Physics, NUS

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(a) Magnetostatics• A magnet can interact with other magnets through the magnetic field. An importantproperty of this magnetic field is given by Gauss's law for magnetic flux.• This magnetic field can interact with moving charges (i.e. electric currents) through the Lorentz force equation.• An electric current produces a magnetic field through the Biot–Savart law.(b) Electrodynamics• A time-varying electric field induces a magnetic field through the Ampere's law.• A time-varying magnetic field induces an electric field through the Faraday's law.We will study each of these in turn between Chapters 7 and 12.

Overview of middle section of PC1432

(a) Magnetic field and force:

magnetic field moving charge or electric current

Lorentz forceexerts force on ...

has a ...Biot–Savart law

(b) A time-varying ... field makes a ... field:

magnetic field electric fieldFaraday’s law

Ampere’s law

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7.1 MagnetismPermanent magnets

A freely suspended bar magnet aligns to the N-S direction. The end that points to the Earth's magnetic north pole is called the north pole; the other end that points to the Earth's magnetic south pole is called the south pole. Opposite poles attract; like poles repel.However unlike electric charges, the north and south poles of the magnet cannot be separated. So far, isolated north and south magnetic monopoles have not been found.Qn: What happens if you keeping smashing up a magnet: can you get to the monopole?

Natural: Lodestone (Fe oxides)source: wikipedia

source: http://www.unitednuclear.com/magnets.htmMagnets come in all shapes and sizes

source: http://www.xionghaimagnets.com/

Fridge magnets N

S

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7.2 Magnetic fieldA magnet creates a magnetic field in space, which we denote by the symbol . This field exerts a magnetic force on other magnets.F

B

• The magnetic field is a vector field (recall: the electric field is also a vector field). The field is described at every point in space by a magnitude and a direction.

• One way to represent a vector field is by using a set of vectors in space:

• The direction of this field is the direction that the north pole of a "test compass" points at that location. This is the same as the direction of the magnetic force on a hypothetical north monopole.• Its magnitude is also called the magnetic flux density or the magnetic induction. Its unit is tesla, abbreviated T (SI unit), or gauss G (cgs unit, 1G = 1x10–4 T).

B

butto

n mag

net

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• The cross-product can also be written explicitly as , where θ is the angle between and .

Bxv

θ⋅⋅ sin|||| Bv

v

B

What force does the magnetic field exert on a moving charge?

BxvqF

⋅=

• Right-hand rule gives the direction of the cross product: let your first (fore) finger point in the direction of the first vector , let yoursecond (middle) finger point in the direction of the second vector , then extend your thumb naturally to get the direction of the cross-product,

v

B

Bxv

• If q is positive, is in the same direction as , otherwise is in the opposite direction.

Bxv

F

F

(a) The direction and magnitude of this force

Experimentally, it was found that the magnetic force is given by

The cross product also defines the right-hand cartesian axes: kjxi ˆˆˆ =is the unit vector in the x-direction, is the unit vector in the y-direction, and is the unit vector in the z-

directioni j k

scalar product vector product

Bv

θ

Bxv

cross product in general:

• Hence magnetic force on a charge element dq is given by BxvdqFd

⋅=

infinitesimal charge

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This may seem very simple, until one realises that according to Newton, motion is relative, and so whether the charge is stationary or moving (and at what speed) depends on one's frame of reference. Therefore according to the equation, the existence and magnitude of the magnetic force depends on the chosen frame of reference. • E.g., if you are moving together with the charge, the charge is at rest in your frame, and so there should be no magnetic force on the charge. Is there a contradiction? • Whose frame of reference is correct - what is the force acting on the charge? This paradox was solved in the early 1900s by Einstein's theory of special relativity: the notion that space and time are not absolute, and so no contradiction exists. For our purpose, it is good enough to know that the equation is always correct in whichever frame of reference in which the force and velocity is measured.

• Another way of thinking about this: can be written as ,which is the product of "velocity" and the "magnetic field component perpendicular to it"; itcan also be written as ,which is the product of "magnetic field" and the "velocity component perpendicular to it".

θ⋅⋅ sin|||| Bv )sin|(||| θ⋅⋅ Bv

)sin|(||| θ⋅⋅ vB

• Therefore, if the charge is not moving, or if it is moving along the direction of , the charge does not experience a magnetic force.

B

Whose frame of reference?

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(b) How large is 1 T?

In SI units, q is in coulombs (C), is in metres per second (m s–1), is in teslas (T), and is in newtons (N). Therefore 1 N = 1 C m s–1 T.• Rearrange the units, 1 T = 1 N (C s–1 m)–1 = 1 N (A m)–1.• Also, in terms of the magnetic flux (Wb is webers, the SI unit for magnetic flux, more on this later) density, 1 T = 1 Wb m–2

v

F

B

Therefore you can think of 1 T as the magnetic field magnitude that produces a force of 1 N on an electric charge of 1 C moving perpendicular to it at a speed of 1 m s–1.

Earth's magnetic field (at the surface) 10–4 TFerrite magnet 0.15 TAlnico magnet 0.15TNeodymium magnet (the most powerful magnet) 1.4 TSuperconducting electromagnet (the most powerful one) 45 TNeutron star’s magnetic field (on its surface) 109 T

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If both electric and magnetic fields are present, the total force on the moving electric charge is simply the vector sum of both the electric force and the magnetic force, which together is called the Lorentz force,

)( BxvEqF

+=

E

B

The Lorentz force

electric force magnetic force

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Example. Magnetic force on a moving proton

A beam of protons (q = 1.602 x 10–19 C) moves at 3.0x105 m s–1 through an uniform magnetic field (2.0 T) directed along the +z-direction. The velocity of the protons at a particular instant is in the xy-plane at an angle of +30º from the +x-direction (anticlockwise, as viewed from the +z-direction):(i) Compute the force on each proton. (ii) Compute the acceleration. (iii) Describe the subsequent motion of these protons.

Step 1. Identify the physics involved.Step 2. Set up the cartesian axes.Step 3. Sketch a diagram, and mark the fields and forces.Step 4. Perform the computation/ integration.

10(Intentionally left blank for the solution.)

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7.3 Magnetic field lines and magnetic fluxBesides the vector representation, the magnetic field can also be represented by field lines in space, just as the case of the electric field. • The local direction of the field is given by the direction of the field line.• The local magnitude of the field is given by the density of these field lines (i.e., how closely spaced they are).

• Uniform magnetic field: indicated by a set of uniformly spaced magnetic field lines all pointing in the same direction.• Non-uniform magnetic field: the region with a higher density of lines has a higher magnetic field than one with lower density of lines.• If magnetic monopoles exist, the field lines will start from the Nmonopole and end at the S monopole.• Since magnetic monopoles have not been found, the magnetic field lines must form continuous unbroken loops. The location on the magnet where the field lines emerge into the “outside world” is the magnetic N pole, and the location where the field lines return into the magnet is the magneticS pole.

B

B

What can you tell from the appearance of the magnetic field lines?

bar magnet

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(a) Magnetic flux through a surface

• Recall that this is analogous to the definition of the electric flux, AdEd E

⋅=Φ

• The dot product here can be written explicitly as θ⋅⋅ cos|||| AdB

where θ is the angle between the and the vectors.B

Ad

• The direction of the vector is perpendicular to the surface element. Its magnitude is the area of that surface element. There are two possible reference directions, or what I call two senses, to choose from: up or down; left or right; front or back. You can choose whichever sense you like, but your results must be treated self-consistently.]

Ad

• The SI unit of ΦB is therefore T m2, which is named the name weber (Wb).

• One way of thinking about this: the amount of flux through an area element is the product of that area and the magnetic field component perpendicular to it.

The magnetic flux element is defined as AdBd B

⋅=Φ

area element

• The dot product is a signed scalar quantity, which means its sense, + or –, is related to your chosen reference direction.

Ad

θ

B

AdBd B

⋅=Φ

is negative.

Ad

θ

B

AdBd B

⋅=Φ

is positive.

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(b) Gauss's law for the magnetic fluxSince we assume that magnetic monopoles do not exist, there cannot be a start or an end to every field line, and so these field lines must form closed loops. Hence the closed surface integral of the magnetic flux must equal zero.

0=⋅=Φ ∫∫surfacesurface

B AdBd

closed surface integrals

Once you know the magnetic flux through each area element, you can get the total magnetic flux through the surface simply by summing over all its area elements. In calculus, this corresponds to an integrationThus the total magnetic flux through any specified surface is then given by the surface integral

∫∫ ⋅=Φ=Φsurfacesurface

BB AdBd

Ad

Ad

Ad

For a closed surface, by convention, the sense of is chosen to be the direction away from the interior bounded by the surface. Thus a positive value for the flux means there is a net flux coming out of the surface.

Ad

direction of surface normal for a closed surface

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Example. Magnetic flux calculations

A flat surface with area 3.0 cm2 is placed in a uniform magnetic field such that the angle between the plane of the surface and the magnetic field is 30º. The magnetic flux through this surface is 0.90 mWb (milliwebers, not meter∙webers!). (a) Compute the magnitude of the magnetic field. [Note: The angle between the plane and its normal is of course 90º, and so the angle between the normal and the magnetic field here is 60º.](b) If the surface is now rotated such that the surface plane is parallel to the magnetic field lines, what is the magnetic flux through the surface?

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7.4 Magnetic force on a current-carrying conductor• Magnetic force on a charge element dq is given by BxvdqFd

⋅=

• By definition, an electric current is the flow of electric charges. The rate at which these charges cross an area is given by v

ddqI ⋅=

BxdIFd

⋅=Thus the magnetic force on a current element is given by

This shows that the direction of the magnetic force is determined by the direction of the conventional current (i.e., the flow of hypothetical positive charges), irregardless of whether the actual current is due to the flow of positive or of negative charges (or even of both).

dIvdq ⋅=⋅we get • This assignment ensures that if dq is positive, the direction of the current is the same as , but if dq is negative, the direction of the current is opposite to .

v

• Multiply both sides by , and let us make this length element into a vector by assigning to it the direction of the current I,

charge per unit length of wirespeed of charges

d

d

v

Direction of the conventional current and actual charge flow:

length of the current element with direction given by that of the conventional current

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∫ ⋅=line

BxdIF

Once you know the magnetic force on a length element, you can find the total magnetic force on the conductor by the line integration

• For a straight section of a conductor carrying current I in a uniform magnetic field, the integration simplifies to:

BxIBxdIFline

⋅=⋅= ∫

This force acts on the moving charges, and tends to push them to one side of the conductor. Since these charges cannot escape from the conductor, the conductor must push back the charges with same force, which means the charges push against the conductor with this force (Newton's action–reaction pair).

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Example. Magnetic force on a conductor in a uniform magnetic field

(a) What is the magnetic force on this U-conductor (the U-part has radius R)?

I

B

Step 1. Identify the physics involved.Step 2. Set up the cartesian axes.Step 3. Sketch a diagram, and mark the fields and forces.Step 4. Look for simplifying symmetry.Step 5. Perform the computation/ integration.

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(b) What is the magnetic force on this current loop (the loop has radius R) where the spacing between the leads is negligible?

I

B

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7.5 Force and torque on a magnetic dipoleA current loop is a magnetic dipole. Each current element on the loop experiences a magnetic force given by . The forces on opposite segments of the loop are in opposite directions. They further have the same magnitude if the magnetic field is uniform. Hence in a uniform magnetic field, the net magnetic force on a current loop is zero.

However, since the two opposing forces do not act along the same line, their torques do not cancel. Hence there is a net magnetic torque that tries to rotate the current loop.

where the magnetic dipole moment is given by

AI⋅=µ

• This net magnetic torque is given by Bx

µ=τ

Here I is the conventional current, is the area vector enclosed by the current loop, and its sense is given by the right-hand screw rule: let the four fingers of the right hand follow the conventional current in the loop, then the thumb gives the direction of and of .

A

µ

A

I

µ

A

BxdIFd

⋅=

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• Orbiting and spinning electrons in atoms also have magnetic dipole moments. These magnetic dipole moment lets the "current loop" interact with the magnetic field, and is the origin of the magnetic properties of materials.

• The magnetic potential energy of the magnetic dipole is then given by

BUB

⋅µ−=

Recall: Electric potential energy EpUE⋅−=

electric dipole moment electric field

UB

Iµ

θ0 18090 270

θ

Rule of thumb: magnetic dipole tries to align with the external magnetic field

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Summary

What you need to be able to do:(a) Compute/ derive the Lorentz force on a moving electrical charge or electrical current.(b) Compute the magnetic flux from given magnetic field.