Chapter 26: Magnetism: Force and Field Magnets

Magnetism Magnetic forces

Magnetism Magnetic field of Earth

Magnetism Magnetic monopoles? Perhaps there exist magnetic charges, just like electric charges. Such an entity would be called a magnetic monopole (having + or magnetic charge). How can you isolate this magnetic charge? Try cutting a bar magnet in half: N S N S Even an individual electron has a magnetic “dipole”! Many searches for magnetic monopoles—the existence of which would explain (within framework of QM) the quantization of electric charge (argument of Dirac) No monopoles have ever been found:

Magnetism Source of magnetic field What is the source of magnetic fields, if not magnetic charge? Answer: electric charge in motion! e.g., current in wire surrounding cylinder (solenoid) produces very similar field to that of bar magnet. Therefore, understanding source of field generated by bar magnet lies in understanding currents at atomic level within bulk matter. Orbits of electrons about nuclei Intrinsic “spin” of electrons (more important effect)

Magnetism Magnetic force: Observations

Magnetism Magnetic force (Lorentz force)

Magnetism Magnetic force (cont’d) Components of the magnetic force

Magnetism B x x x x x x ® ® ® ® ® ­ ­ ­ ­ ­ ­ ­ ­ v ´ q F F = 0 Magnetic force (cont’d) Magnetic force F x x x x x x v B q ® ® ® ® ® F = 0 ­ ­ ­ ­ ­ ­ ­ ­ ´

Magnetism Magnetic force (cont’d) Units of magnetic field

Magnetism Magnetic force vs. electric force

Magnetic Field Lines and Flux

Magnetic Field Lines and Flux

Magnetic Field Lines and Flux Magnetic field lines (cont’d)

Electric Field Lines of an Electric Dipole Magnetic Field Lines and Flux Magnetic field lines (cont’d) Electric Field Lines of an Electric Dipole Magnetic Field Lines of a bar magnet

Magnetic Field Lines and Flux Magnetic field lines (cont’d)

Magnetic Field Lines and Flux Magnetic field lines (cont’d)

Magnetic Field Lines and Flux Magnetic flux magnetic flux through a surface Area A B  B B

Magnetic Field Lines and Flux Magnetic flux (cont’d) Units: A=C/s, T=N/[C(m/s)] -> Tm2=Nm/[C/s]=Nm/A Gauss’s law for magnetism No magnetic monopole has been observed!

Motion of Charged Particles in a Magnetic Field Case 1: Velocity perpendicular to magnetic field υ perpendicular to B The particle moves at constant speed υ in a circle in the plane perpendicular to B. F/m = a provides the acceleration to the center, so v R F B x

Motion of Charged Particles in a Magnetic Field Case 1: Velocity perpendicular to magnetic field

Motion of Charged Particles in a Magnetic Field Case 1: Velocity perpendicular to magnetic field Velocity selector

Motion of Charged Particles in a Magnetic Field Case 1: Velocity perpendicular to magnetic field Mass spectrometer

Motion of Charged Particles in a Magnetic Field Case 1: Velocity perpendicular to magnetic field Mass spectrometer

Motion of Charged Particles in a Magnetic Field Case 1: Velocity perpendicular to magnetic field Mass spectrometer

Motion of Charged Particles in a Magnetic Field Case 1: Velocity perpendicular to magnetic field Mass spectrometer

Motion of Charged Particles in a Magnetic Field Case 2: General case (v at any angle to B)

Motion of Charged Particles in a Magnetic Field Case 2: General case (cont’d) Since the magnetic field does not exert force on a charge that travels in its direction, the component of velocity in the magnetic field direction does not change.

Magnetic Force on a Current-Carrying Conductor Magnetic force on a current (straight wire)

Magnetic Force on a Current-Carrying Conductor Magnetic force on a current (straight wire) (cont’d)

Magnetic Force on a Current-Carrying Conductor Magnetic force on a current (curved wire)

Magnetic force on a current: Example1 Magnetic Force on a Current-Carrying Conductor Magnetic force on a current: Example1

Magnetic force on a current: Example1 (cont’d) Magnetic Force on a Current-Carrying Conductor Magnetic force on a current: Example1 (cont’d)

Magnetic force on a current: Example1 (cont’d) Magnetic Force on a Current-Carrying Conductor Magnetic force on a current: Example1 (cont’d)

Magnetic force on a current: Example1 (cont’d) Magnetic Force on a Current-Carrying Conductor Magnetic force on a current: Example1 (cont’d)

Magnetic force on a current: Example1 (cont’d) Magnetic Force on a Current-Carrying Conductor Magnetic force on a current: Example1 (cont’d)

Magnetic force on a current: Example2 Magnetic Force on a Current-Carrying Conductor Magnetic force on a current: Example2

Magnetic force on a current: Example2 (cont’d) Magnetic Force on a Current-Carrying Conductor Magnetic force on a current: Example2 (cont’d)

Magnetic force on a current: Example2 (cont’d) Magnetic Force on a Current-Carrying Conductor Magnetic force on a current: Example2 (cont’d)

Plane of loop is parallel to the magnetic field Force and Torque on a Current Loop Plane of loop is parallel to the magnetic field

Plane of loop : general case Force and Torque on a Current Loop Plane of loop : general case if q=90o

Plane of loop and magnetic moment Force and Torque on a Current Loop Plane of loop and magnetic moment

Plane of loop : magnetic moment (cont’d) Force and Torque on a Current Loop Plane of loop : magnetic moment (cont’d) The same magnetic dipole moment formulae work for any shape of planar loop. Any such loop can be filled by a rectangular mesh as in the sketch. Each sub-loop is made to carry the current NI. You will now see that all the interior wires have zero current and are of no consequence. Nevertheless, each sub-loop contributes to μ in proportion to its area.

Plane of loop : magnetic moment (cont’d) Force and Torque on a Current Loop Plane of loop : magnetic moment (cont’d)

Potential energy of a magnetic dipole Force and Torque on a Current Loop Potential energy of a magnetic dipole Work done by the torque when the magnetic moment is rotated by df : In analogy to the case of an electric dipole in Chapter 22, we define a potential energy: Potential energy of a magnetic dipole at angle f to a magnetic field

Applications Galvanometer We have seen that a magnet can exert a torque on a loop of current – aligns the loop’s “dipole moment” with the field. In this picture the loop (and hence the needle) wants to rotate clockwise The spring produces a torque in the opposite direction The needle will sit at its equilibrium position Current increased μ = I • Area increases Torque from B increases Angle of needle increases Current decreased μ decreases Torque from B decreases Angle of needle decreases

Applications Slightly tip the loop Restoring force from the magnetic Motor Slightly tip the loop Restoring force from the magnetic torque Oscillations Now turn the current off, just as the loop’s μ is aligned with B Loop “coasts” around until its μ is ~antialigned with B Turn current back on Magnetic torque gives another kick to the loop Continuous rotation in steady state

Applications Even better Motor (cont’d) Even better Have the current change directions every half rotation Torque acts the entire time Two ways to change current in loop: Use a fixed voltage, but change the circuit (e.g., break connection every half cycle  DC motors 2. Keep the current fixed, oscillate the source voltage AC motors VS I t

Applications Motor (cont’d) flip the current direction

Applications - - - + + + charges accumulate (in this case electrons) Hall effect - - - + + + Measuring Hall voltage (Hall emf) In a steady state qEH =qvdB Charges move sideways until the Hall field EH grows to balance the force due to the magnetic field: n can be measured

Applications Electromagnetic rail gun

Exercises Exercise 1 If a proton moves in a circle of radius 21 cm perpendicular to a B field of 0.4 T, what is the speed of the proton and the frequency of motion? 1 v r x 2

Exercises Exercise 2 Example of the force on a fast moving proton due to the earth’s magnetic field. (Already we know we can neglect gravity, but can we neglect magnetism?) Let v = 107 m/s moving North. What is the direction and magnitude of F? Take B = 0.5x10-4 T and v B to get maximum effect. (a very fast-moving proton) B N F v vxB is into the paper (west). Check with globe

Magnetic Field of a Moving Charge Magnetic field produced by a moving charge Note the factor of μ0 /4π the constant of proportionality needed just as 1/(4πε0) is needed in electrostatics.

Magnetic Field of a Current Element Magnetic field produced by a current element ds For an element ds of a conductor carrying a current I there are n A ds charges with drift velocity υd (using priciple of superposition). number of charge q ds ds

Magnetic Field of a Current Element Biot-Savart law ds Note: ds is dL in the textbook. ds Note that ds is in the direction of I, but has a magnitude which is ds the length of wire considered. Deduced by Biot and Savart c. 1825 from experiments with coils

r Magnetic Field of a Current Element  dB P Biot-Savart law (cont’d)  dB P r The magnitude of the field dB is: I  ds Total magnetic field at P is found by summing over all the current elements ds in the wire.

Magnetic Field of a Straight Current Carrying Conductor A straight wire of length L A thin straight wire of length L carries constant current I . Calculate the total B field at P.  P x ds r I R y  dB So the magnitude of dB is given by:

Magnetic Field of a Straight Current Carrying Conductor A straight wire of length L (cont’d)  P x ds r I R y  dB

Magnetic Field of a Straight Current Carrying Conductor A straight wire of length L (cont’d)  P x ds r I R y  dB In the limit (L/R) →∞ Magnetic field by a long straight wire

Magnetic Field of a Straight Current Carrying Conductor A straight wire of length L (cont’d) B B B I B

Magnetic Field of a Straight Current Carrying Conductor Example: A long straight wire Iron filings

Note: B field at the centre of a loop, =2 Magnetic Field of a Current Element Example Calculate the magnetic field at point O due to the wire segment shown. The wire carries uniform current I, and consists of two straight segments and a circular arc of radius R that subtends angle . A´ A ds C´ The magnetic field due to segments A´A and CC´ is zero because ds is parallel to along these paths. I  C R Along path AC, ds and are perpendicular. O Note: B field at the centre of a loop, =2

Force Between Parallel Conductors Two parallel wires At a distance a from the wire with current I1 the magnetic field due to the wire is given by

Force Between Parallel Conductors Two parallel wires (cont’d) Parallel conductors carrying current in the same direction attract each other. Parallel conductors carrying currents in opposite directions repel each other.

Force Between Parallel Conductors Definition of ampere The chosen definition is that for a = L = 1m, The ampere is made to be such that F2 = 2×10−7 N when I1=I2=1 ampere  This choice does two things (1) it makes the ampere (and also the volt) have very convenient magnitudes for every day life and (2) it fixes the size of μ0 = 4π×10−7. Note ε0 = 1/(μ0c2). All the other units follow almost automatically.

Magnetic Field of a Circular Current Loop Magnetic field produced by a loop current Use to find B field at the center of a loop of wire. R I Loop of wire lying in a plane. It has radius R and total current I flowing in it. First find is a vector coming out of the paper at the same angle anywhere on the circle. The angle is constant. Magnitude of B field at center of loop. Direction is out of paper. R i

Magnetic Field of a Circular Current Loop Example 1: Loop of wire of radius R = 5 cm and current I = 10 A. What is B at the center? Magnitude and direction I Direction is out of the page.

Magnetic Field of a Circular Current Loop Example 2: What is the B field at the center of a segment or circular arc of wire? 0 R I Total length of arc is S. P where S is the arc length S =R0 0 is in radians (not degrees) Why is the contribution to the B field at P equal to zero from the straight section of wire?

Ampere’s Law Previously from the Biot-Savart’s law we had Ampere’s Law Ampere’s law : A circular path Consider any circular path of radius R centered on the wire carrying current I. Evaluate the scalar product B·ds around this path. Note that B and ds are parallel at all points along the path. Also the magnitude of B is constant on this path. So the sum of all the B·ds terms around the circle is Previously from the Biot-Savart’s law we had Ampere’s Law On substitution for B

Ampere’s Law For any path which encloses the wire z Ampere’s Law ^ k Ampere’s law : A general path ^ y r ^ x q q Let us look at the integral along any shape of closed path in 3D. The most general ds is Where unit vectors are used for the radial r and the tangential directions q and for z along the wire k. In this system we have ^ ^ ^ tangential component of ds For any path which encloses the wire For any path which does not enclose the wire

Ampere’s Law Ampere’s law : This law holds for an arbitrary closed path that is threaded by a steady current. I is the total current that passes through a surface bounded by the closed path.

Ampere’s Law Electric Field General: Coulomb’s Law Electric field vs. magnetic field Electric Field General: Coulomb’s Law High symmetry: Gauss’s Law Magnetic Field General: Biot-Savart Law High Symmetry: Ampère’s Law

Applications of Ampere’s Law Magnetic field by a long cylindrical conductor A long straight wire of radius R carries a steady current I that is uniformly distributed through the cross-section of the wire. Outside R. In region where r < R choose a circle of radius r centered on the wire as a path of integration. Along this path, B is again constant in magnitude and is always parallel to the path. Now Itot ≠ I. However, current is uniform over the cross-section of the wire. Fraction of the current I enclosed by the circle of radius r < R equals the ratio of the area of the circle of radius r and the cross section of the wire R2.

Applications of Ampere’s Law Magnetic field by a long cylindrical conductor B r R

Applications of Ampere’s Law Magnetic field by a circular current Consider circular current carrying loop. Calculate B field at point P, a dist x from the centre of the loop on the axis of the loop. Ids ds Again in this case vector I ds is tangent to loopand perp to vector r from current element to point P. dB is in direction shown, perp to vectors r and I ds. Magnitude dB is: ds ds

Applications of Ampere’s Law Magnetic field by a circular current (cont’d) ds ds Ids Integrate around loop, all components of dB perp to axis (e.g. dBy). integrate to zero. Only dBx , the components parallel to axis contribute. ds Field due to entire loop obtained by integrating:

B on the axis of a current loop Applications of Ampere’s Law Magnetic field by a circular current (cont’d) But I, R and x are constant ds ds Ids B on the axis of a current loop

Applications of Ampere’s Law Magnetic field by a circular current (cont’d) Limits: x 0 x >>R Compare case of electric field on axis of electric dipole far from dipole vs.

Applications of Ampere’s Law Magnetic field by a solenoid When the coils of the solenoid are closely spaced, each turn can be regarded as a circular loop, and the net magnetic field is the vector sum of the magnetic field for each loop. This produces a magnetic field that is approximately constant inside the solenoid, and nearly zero outside the solenoid. I

Applications of Ampere’s Law Magnetic field by a solenoid (cont’d) The ideal solenoid is approached when the coils are very close together and the length of the solenoid is much greater than its radius. Then we can approximate the magnetic field as constant inside and zero outside the solenoid. I

Applications of Ampere’s Law Magnetic field by a solenoid (cont’d) Use Ampère’s Law to find B inside an ideal solenoid.

Applications of Ampere’s Law Magnetic field by a toroid A toroid can be considered as a solenoid “bent” into a circle as shown. We can apply Ampère’s law along the circular path inside the toroid. N is the number of loops in the toroid, and I is the current in each loop

Exercises Problem 1 The wire semicircles shown in Fig. have radii a and b. Calculate the net magnetic field that the current in the wires produces at point P. I Since point P is located at a symmetric position with respect to the two straight sections where the current I moves (anti)parallel to the radial direction. So there is no contributions from these segments. The contribution from the semicircle of radius a is a half of that from a complete circle of the same radius: Similarly the contribution from the semicircle of radius b is: From principle of superposition, the net magnetic field at point P is: I P

Exercises Long, straight conductors with square cross sections and Problem 2 Long, straight conductors with square cross sections and each carrying current I are laid side-by-side to form an infinite current sheet. The conductors lie in the xy-plane, are parallel to the y-axis and carry current in the +y direction. There are n conductors per unit length measured along the x-axis. (a) What are the magnitude and direction of the magnetic a distance a below the current sheets? (b) What are the magnitude and direction of the magnetic field a distance a above the current sheet? x z y

Exercises B Below sheet, all the magnetic Problem 2 (cont’d) B Below sheet, all the magnetic field contributions from different wires add up to produce a magnetic field that points in the positive x-direction. Components in the z-direction cancel. Using Ampere’s law, where we use the fact that the field is anti- symmetric above and below the current sheets, and that the legs of the path perpendicular provide nothing to the integral. So, at a distance a beneath the sheet the magnetic field is: b) The field has the same magnitude above the sheet, but points in the negative x-direction. L B