Chapter 20: Magnetic field and forces What will we learn in this chapter? What is magnetism? Magnetic fields and forces Motion of charges in a magnetic field Mass spectrometers Magnetic forces on conductors Magnetic forces on a loop Magnetic field of a straight conductor Magnetic forces between parallel conductors Magnetic field calculations iron filings line up tangent to the field lines
Why is magnetism important? Technological relevance: Most common electric motor technology relies on magnetism Generators require magnetism to produce electricity All disk drives are based on magnetic materials Two types of magnets: Permanent magnets: materials, which intrinsically have magnetic moments. Electromagnets: when charges move, a magnetic field is induced. Monopoles? Typical behavior of magnets: Unlike poles attract Same poles repel What happens when we cut a magnet into two pieces? S N S S N N + There are no magnetic monopoles. Note that this would have huge implications in theoretical physics.
Earth: a huge magnet The reason why a compass points to the north pole is because the north pole is actually a magnetic south pole. Earth s axis is slightly offset from the magnetic axis. Earth behaves like a large bar magnet. The field is caused by currents in the planet s core. It periodically flips (10 6 years ). Birds and bacteria use this magnetic field to orient themselves. Brief history of magnetism Permanent magnets (500 BC): Fragments of magnetized iron are found near the city of Magnesia (today Manisa, TK). When suspended from a string, the magnetized iron would point north. Electromagnetism (1819): Hans Christian Ørsted (DK) observes that a compass needle is deflected when placed next to a current-carrying wire. I = 0 I > 0 I < 0
Electric vs magnetic fields Electric field: A charge distribution at rest creates an electric field E at all points in the surrounding space. The electric field exerts a force F = qe on any other charge q present in the field. Magnetic field: A permanent magnet, a moving charge, or a current create a magnetic field B at all points in the surrounding space. The magnetic field B exerts a force F on any other moving charge or current present in the field. Note: Like the electric field, the magnetic field is a vector field. Magnetic field lines As for electric fields, we can draw magnetic field lines. The magnetic field vector B is tangent to the field lines. The more dense the lines the stronger the field. At each point, the field lines point in the same direction as a compass would. Because the field is unique, field lines never cross. Convention: Magnetic field lines point away from N poles and toward S poles.
Some example cases C-shaped magnet Loop and coil The field is almost uniform between the poles. similar field lines to a bar magnet... Straight wire with current Visualizing field lines Iron shavings can be used to visualize field lines. field lines go from N to S field lines go from N to S
Magnetic force Magnitude of the magnetic force: When a charged particle moves with velocity v in a magnetic field B, the magnitude F of the force exerted on it is F = q v B = q vb sin(φ) Here q is the magnitude of the charge and φ is the angle measured from the direction of v to the direction of B. right index right thumb F =0 right middle finger F = q v B = q vb sin(φ) If v B the force is maximal. Note the difference to electric fields! Magnetic force contd. One can equivalently also write F = q vb. Thus the direction of the force is not specified completely. Solution: introduce a right-hand rule Right-hand rule for magnetic forces (positive charge): 1.! Draw the vectors v and B with their tails together. 2.! Imagine turning v in the plane containing v and B until it points in the! direction of B. Turn it trough the smaller of the two possible angles. 3.! The force then acts along a line perpendicular to the plane! containing v and B. Using your right hand, curl your fingers around! this line in the same direction (clockwise/counterclockwise) that! you turned v. Your thumb now points in the direction on of the! force F on a positive charge. If the charge is negative, the force points opposite the direction given by the right-hand rule.
Right-hand rule in pictures Right-hand rule for positive charges: Note: Mathematically, the force is actually given by the cross product (or curl ) between the field and the velocity,. F = q v B Negative charges (left hand) Different charges will move in opposite directions if they move in the same direction in a magnetic field.
Units The units of B must be the same as the units of F/qv, i.e., N/(Cm/s). Because 1A = 1C/s it follows [B] = 1 N/(Am). The unit for a magnetic field is called Tesla:!! 1 Tesla = 1 T = 1 N/(Am) The cgs unit (often found) of B is gauss (1G = 10-4 T). Typical field strenghts: Earth s magnetic field is about 0.5G. Permanent magnet or microwave oven: 0.1T. MRI machines have about 1.5T fields (erase credit cards). Strongest steady field in laboratories are around 45-60T. How do MRI machines work? Superconductors generate a huge magnetic field of 1.5T. By pulsing the field, the magnetic moment of the hydrogen atoms in the tissue flip back and forth. Different tissues have different amounts of hydrogen. Their flipping is detected and the figure is formed. How is the signal of the atoms detected? Use use tiny superconducting pick-up coils which can sense the faintest fields.
Velocity selector Many applications use beams of charged particles. Common particle beam sources produce a range of particle velocities. How can we select a given particle speed? Send the particles trough a capacitor (E field points left) and a magnetic field (B field in plane). When the fields are tuned properly F E = F B qe = qvb It follows: v = E/B Only particles with a speed of E/B will fly trough the capacitor. Thomson s experiment: in 1894 Sir J. J. Thomson used this method to determine the ratio e/me. He noted that it was independent of the used material, i.e., he discovered subatomic particles. Later Millikan computed e and thus the electron mass was determined. particle source Motion of charges in magnetic fields When a charged particle moves in a magnetic field, B, v, and F are always perpendicular to each other. The magnitude of the magnetic force, qvb, does not change. Only the direction with the aforementioned constraint. It follows that the charged particles follow a circular path on a plane perpendicular to the field B. The radial acceleration is v 2 /R. Using Newton s 2. law: F = q vb = m v2 R The radius R of the circle is: R = mv q B Note: if q < 0 the particle moves clockwise.
Motion in a magnetic field contd. Angular velocity of a particle: ω = q B m ω =2πf Note: The frequency is called the cyclotron frequency. It is independent of the radius R. This is used in particle accelerators (CERN, microwave oven, radar) where particles are given a boost twice each revolution. Helical motion If a particles moves in a uniform magnetic field and the initial velocity is not perpendicular to the field, there is a parallel velocity component. Because there is no force parallel to the field, this velocity component is constant. The particle moves in a helix of radius R: R = mv q B This is the reason for aurora borealis: Particles from the sun are trapped at the poles and collide with air molecules emitting light.
Mass spectrometers Recall that a velocity selector can select particles with an exact velocity v = E/B. Ions are sent trough slits S1 and S2 to make a narrow beam. The ions are sent into a region where only a given magnetic field (no E field) perpendicular to the figure. It follows: R = mv q B If we know how many electrons an ion has lost, we can determine m. A counter then counts the number of hits and a database search determines the relative frequencies of the particles in the unknown material. This is how isotopes were discovered. velocity selector Mass spectrometers contd. Selected applications: Isotope count ( 14 C dating) Trace gas analysis Pharmacokinetics Protein characterization Space exploration typical data set Often shown in TV shows (CSI). Note that the process never is as fast as it seems on TV
Magnetic force on a conductor w. current What makes an electric motor work? How can we generate energy from e.g., water, wind, gas, fission? The forces that make motors turn are forces a magnetic field exerts on a current-carrying conductor. Calculation of the force: Assume positive charges in a wire of length l and cross section A in a in-plane field B. The force on a single charge is F = q vb sin(φ) Since the drift velocity vd, field B and force F are perpendicular F = q v d B. The total force can be computed by studying the total charge moving in the conductor Magnetic force calculation contd. The time needed for a charge Q to move from one end of the conductor to the other is t = l The total charge flowing during this time trough the wire is Q = I t Use now F = Qv d B : v d F = Qv d B =(I t)(l/ t)b = IlB Magnetic force on a current-carrying conductor: The magnetic force on a segment of length l carrying a current I in a field B is F = IlB = IlB sin(φ) The force is always perpendicular to both the conductor and the field, with the direction determined by the right-hand rule.
Reversing the field or current When the current in the wire or the field B are reversed the force changes direction (use the RH rule ). What happens when the charges are negative? Nothing! The sign of the velocity and the charge cancel. Note: moving the wire in a field produces a current (generator). Force and torque on a current loop For simplicity, assume the loop is rectangular. Due to symmetry considerations, the force on the loop is zero, but there is a net torque. On the long side of the loop there are forces F and F acting on the wire with F = IaB Along the short sides the force makes an angle 90 φ with the field, hence F = IbB sin(90 φ) =IbB cos(φ) (along the y axis).
Force & torque on a current loop contd. Since the magnitudes of the forces on opposing sides of the loops are the same, but with different signs, they cancel. The forces along the y axis act along the same line, i.e., no net torque. One can compute the torque to find φ = 90 φ =0 τ =(IaB)(b sin φ) When φ =0 we have a stable equilibrium because the torque is maximal torque zero. When φ = 180 we have an unstable equilibrium position. Since the area of the loop is A = ab, we can write τ = IAB sin φ zero torque Force & torque on a current loop contd. Changing the shape of the loop: Any arbitrarily-shaped planar loop can be approximated by a set of rectangular loops. It follows that τ = IAB sin φ still holds with A the area of the loop. Multiple loops: The effect of each loop adds up. Thus for N loops τ Nτ
Force & torque on a current loop summary Torque on a current-carrying loop: When a conducting loop of area A carries a current I in a uniform magnetic field B, the torque exerted on the loop is τ = IAB sin φ where φ is the angle between the normal to the loop and the field B. For multiple loops (solenoid with N loops) τ = NIAB sin φ. The torque tends to rotate the loop in the direction of decreasing angle (stable equilibrium). The product IA is called the magnetic moment of the loop: µ = IA For a solenoid with N windings µ = NIA. Note: A solenoid rotates in a field and become parallel to the field (compass). Direct-current (dc) motor Now that we know how to generate torque on a wire, we can build a motor Center: rotor of soft steel which rotates. Copper wires are embedded in the rotor. Brushes (graphite) pass current onto the rotor via the commutator which switches the currents on and off. In the field coils F and F a steady magnetic field is set up.
Direct-current motor contd. The brushes transmit current onto contacts 1 and 1 to coil 1, which is oriented to have maximal torque. When the brushes are between contacts, inertia keeps the motor running. After a rotation, the brushes touch 1 and 1, but with opposed polarity, hence the torque is still counterclockwise. Magnetic field of a long straight conductor So far: We have only studied forces caused by magnetic fields. We did not worry how these fields are produced. Now: Study some simple cases, such as a long straight wire carrying a current I. The field depends on the distance r from the wire and the current I. Along a field line B has the same magnitude. RH rule: the thumb points in the direction of the field.
Application: coaxial cables Goal: Shield sensitive electromagnetic signals from noise. Avoid signal leakage. Solution: Place a wire inside a hollow conducting tube. If currents in both conductors have the same magnitude and are opposed, fields cancel. Do not introduce kinks! Experiment at home with the TV Magnetic field of a straight conductor contd. Magnetic field of a long straight wire: The magnetic field B produced by a long straight wire carrying a current I at a distance r has a magnitude µ 0 B = µ 0 I 2π r is the permeability of vacuum. Its numerical value depends on the used units. For SI units µ 0 =4π 10 7 Tm/A = 4π 10 7 N/A 2 Note: Do not confuse the permeability µ 0 with the magnetic moment µ! Remember: Magnetic field lines encircle the current that acts as their source. In contrast, electric field lines originate at charges which act as their source.
Force between parallel conductors Motivation: The problem appears in many applications. Significant for the definition of the Ampere. Setup: Two wires with currents I and I in the same direction at a distance r. The magnetic field of the lower wire exerts a force on the upper wire and vice versa. Note: If the wires had currents in the opposite directions, they would repel each other. Force between parallel conductors contd. Determination of the force per length: The magnitude of the B vector at points on the upper conductor is B = µ 0I 2πr Use now that the force on a wire of length l is given by F = IlB to obtain ( ) F = I Bl = I µ0 I l = µ 0II l 2πr 2πr The force per unit length is thus F l = µ 0II 2πr Using the right-hand rule it follows that the wires attract (repel) each other when current flows in the same (opposite) direction.
Example: superconducting wires What is the force of two superconducting cables 4.5mm apart which each carry 15000A? Solution: F l Note: = µ 0II 2πr =1.0 104 N/m For 1m of wire, this would be equivalent to one ton (!). This is why the structural design of high-intensity magnets is crucial. FYI: the definition of the ampere The forces that two straight parallel conductors exert on each other form the basis for the official SI definition of the ampere: Definition of 1 ampere: One ampere is that unvarying current which, if present in each of two parallel conductors of infinite length and 1 meter apart in vacuum causes a force of exactly 2. 10-7 N per meter length on each conductor. Note: This SI definition agrees with the definition of the constant. There is a proposal to redefine the ampere as a number of electrons per second. µ 0
Current loops and solenoids As we have seen in the case of electric motors, many applications require current loops and coils made from many circular loops. Magnetic field in the center of a loop: B = µ 0I 2R If we have a coil of N loops Note: B NB The field lines (determined by the RH rule) are not circular. Note the similarity with the equation for a straight wire. Magnetic field of a solenoid A solenoid is basically a multi-wind coil where the length L is much larger than the radius R of the windings. The field is strongest in the center and almost uniform along the x-axis. The magnetic field depends on the number of turns per unit length n = N/L. At the center we find B = µ 0 ni Remember: This expression is only valid for the case where L is much larger than R.
Special case: toroidal solenoids When the windings are tied very closely, the field is enclosed almost entirely in the area enclosed. If there are N turns then the field B at a distance r from the center of the toroid is B = µ 0NI 2πr If the radial thickness of the core compared to the radius of the toroid is small, then the field is uniform and B = µ 0 ni with n = N/2πr. Law of Biot and Savart So far: Only results for the B-field with different geometries. These results stem from the law of Biot & Savart. Description: The law of Biot & Savart gives the magnetic field B produced by a current I in a tiny segment of conductor of length l. The vector l is the line segment in the direction of the current. The segment is called a source point, the place we want the field at at a distance r, the field point. Law of Biot & Savart: The magnitude of the magnetic field B due to a segment of conductor with length l carrying a current I is B = µ 0 I l sin θ 4π r 2
Law of Biot and Savart contd. zero field cross-section view maximal field Note: the right-hand rule applies. Law of Biot and Savart contd. How can the magnetic field for a large object be computed? Solution: Represent the conductor of length l as a superposition of tiny wire segments l. The total field B is then the vector sum over all B s. Note: mathematically one would perform a line integral along a given path. This problem can usually only be solved analytically for simple cases. Here we sketch a simple example (conducting ring) where we can avoid cumbersome integrations.
Field of a conducting ring revisited Start from Biot & Savarts law: B = µ 0 I l sin θ 4π r 2 We can represent the loop as a sum of tiny ring segments l i. Because of the ring geometry l B R θ = 90 It follows: B = µ 0I 4πR 2 ( l 1 + l 2 + l 3 +...) Use the fact that the sum over line segments is the circumference of the loop, i.e., l 1 + l 2 + l 3 +... =2πR We obtain: B = µ 0I 2R Ampere s law Ampere s law provides an alternative formulation of the relationship between a magnetic field and its source. It is analogous to Gauss law. Ampere s law is useful when the problem has some symmetry which can be exploited. closed loop parallel component Ampere s law: When a path is made up of a series of segments s and when the path links conductors carrying a total current I encl B s = µ 0 I encl Note: If there are more conductors in the path, then the field is the vector sum of the individual contributions, whereas the current is the algebraic sum. In general: Bd s = µ 0 I encl
Example of Ampere s law: long straight wire The magnetic field lines produced by a long straight wire with a current are circles. In this case the parallel component of the magnetic field is always tangential to the circle. Ampere s law: B s = B(2πR) =µ 0 I encl Result: B = µ 0I 2πR Magnetic materials In general, magnetic materials can be classified in different categories: Paramagnets (iron oxide, uranium, platinum, tungsten, oxygen, ) Diamagnets (bismuth, copper, carbon, water, silver, lead, ) Ferromagnets (cobalt, iron, nickel, gadolinium, dysprosium, ) Antiferromagnets (chromium, iron-manganese, nickel oxide, ) Ferrimagnets (magnetite, yttrium-iron-garnet, ) Spin glasses (iron-gold alloys, diluted LiHo, ) Most common are the first 3: paramagnets, diamagnets and ferromagnets ( stick to fridge ).
Magnetic materials contd. Paramagnetism: All equations shown are valid for paramagnets. The strength of the field is quantified with the permeability Km which typically is 1.00000002 to 1.0004 for typical materials. To take this into account, replace µ 0 µ = K m µ 0. Diamagnetism: In this case Km is smaller than 1, i.e., a deflection due to a field would happen in the opposite direction. Ferromagnetism: Km is typically 10 3 to 10 4 due to cooperative effects between domains. Some materials retain the magnetization and thus become permanent magnets. B =0 B>0