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How Do Magnets Behave? • A Magnet has two poles: “North” and “South”. Example: The Earth. • For two magnets: –Unlike poles attract; like poles repel. • The North Pole of a magnet points North (towards the Arctic) by definition. •Where is Earth’s South Pole? –In Canada. •Where is Earth’s North Pole? –In Antarctica. •Magnetism acts through a Magnetic Field (a “ B-field ”) (S.I. Unit: “Tesla” (T)) How can we Visualize Magnetic Fields? • Magnetic Field Lines – Lines start at North Pole and end at South Pole. – Lines must have a start and end; a North Pole cannot exist without a South Pole. – Direction of a line at a point is in the direction of the B-field. – Density of lines is proportional to the magnetic field strength. – Patterns Similar to E-field lines can be found: • Magnetic Dipole • Repulsive poles • Magnetic Quadrupole. How is Magnetism caused? • Magnetism is – caused by moving charges. – felt by moving charges. • The charges must be moving to cause or be affected by magnetism. • A stationary charge will cause and feel an E-field. • A moving charge will cause and feel an E-field and a B-field. • Electric Currents (moving charges) – Cause B-Fields (Hans Oersted, 1820) – Feel B-Fields Questions? • But isn’t a charge’s motion relative to an observer? – The magnetic and electric fields observed depend on your reference frame. • What about magnets? Where are their moving charges? B-Fields caused by Currents • A current causes a B-field that is perpendicular to current flow. • Point your right thumb in the direction of the current; your fingers will curl in the direction of the B-field. (“Right Hand Rule #1”). Forces between Current Carrying Wires. Wires with currents in the same direction attract. The opposite B-fields between the wires are as two unlike poles. Wires with currents in opposite directions repel. The B-fields between the wires are in the same direction and are as two like poles. Magnitude and Direction of Force on a Current Carrying Wire due to a B-Field • Magnitude: For a Wire having current I and length (l ) making an angle (θ) with the B-field: FB = I l B sin θ • Direction: – Right Hand Rule #2 – Thumb in direction of Current. – Fingers in direction of B-Field. – Palm points to direction of Force. NOTE: FB is PERPENDICULAR TO BOTH I and B. Examples 1. Verify Right Hand Rule #2 For these wires. 2. Suppose the current in a wire is in the same direction as the B-Field. In what direction is the Force? Answer: The Force is zero because sin θ is zero. Magnitude and Direction of Force on a Charged Particle due to a B-Field • Suppose we follow one charged (+q) particle in a current: – Particle has velocity (v) in the direction of the current. – In time (Δt) the charge covers a distance: l= vΔt • Thinking of one charged particle as a small current, the force is FB = IlB sinθ = (q/Δt) (vΔt) B sinθ • Magnitude: FB = qvBsinθ • Direction: Use RHR#2 with v instead of I Examples 1. A wire of length l =.12 m and current I =30A makes an angle of 60o with a B-field having a magnitude of .90 T in the x-direction as shown. What is the Force on the wire? I θ B Examples (cont’d) 2. A square wire loop has a mass m = 0.5kg and a current I=10.0A. The loop hangs from a spring scale measuring in newtons. If the bottom of the loop is in a uniform B-field of 0.5T coming out of the page, what is the reading on the spring scale? Examples (cont’d) 3. A positron (charge: +e) enters a region of uniform magnetic field of B = .10T directed into the page. The positron’s initial velocity is v =106m/s in the +x direction. a. What path does the particle follow? b. What is the rate at which work is done on the particle by the B-field? c. How would the path change if the particle were an electron? d. How is this scenario used by particle physicists? Examples (cont’d) 4. Mass Spectrometer: Using Applied Electric and Magnetic Fields to find a particle’s mass (see pg. 642 of text): Northern Lights (Example 5) Magnetic Field due to a Long Wire (WITHOUT PROOF) Magnetic Permeability of Free Space: Force Between Two Wires Carrying Current L d Force on wire 1 by wire 2: F1 = I1 B2 (to the right) Force on wire 2 by wire 1: F2 = I2 B1 (to the left) (So the wires attract) Magnetic Fields of Loops, Coils, and Solenoids (WITHOUT PROOF) • For a wire loop of radius (r) carrying a current (I), the B-field at the center of the loop is: B = (μ0 I)/2r • For a coil of N-loops of radius (r): B = (μ0 N I)/2r (WHY?) • For a solenoid of N-loops and length (L): B = (μ0 N I)/L (WHY?) or: B = μ0 n I ; n = N / L Convenient Alternate RHR1 for Loops, Coils, and Solenoids Curl your fingers in the direction of the current then thumb points in the direction of the B-field. How do Magnetic Materials cause B-fields? • Electrons in atoms have “spin”; an intrinsic fixed quantity of angular momentum. – Spin can be “up” (+) or “down” (-) – Unpaired electrons in atom give the atom a net spin. • A net spin (rotational vector) – acts an effective current associated with the atom. – The spin and B-field of current are the same. • If all atoms have their spins in same direction – An effective BOUND surface current exists and …. – the material is magnetic Magnetic Materials (cont’d) Nonmagnetic Material (spins randomly directed) Permanent Magnet (spins aligned) Ferromagnetism • A ferromagnetic material has magnetic domains, regions in which spins line up, but domains cancel on average. • Applying an external Bfield causes the domains to line up and the sample becomes a magnet. (Example: electromagnet) Magnetic Materials (cont’d) • Permanent Magnet- Spins aligned on neighboring atoms (“ordered”), appears magnetic. • Nonmagnetic – Spins randomly distributed; B-fields cancel out on average. • Antiferromagnet – Spins on neighboring atoms aligned in opposite directions (“ordered”); B-fields cancel locally. A Magnetic Flux • Magnetic Field Strength (B): – S.I. Unit: Tesla (T) – Proportional to density of field lines • Magnetic Flux (Φm) – Φm = (B┴ × A) = BAcos(θ) – Proportional to number of field lines through an area. – S.I. Unit: Weber (Wb) (1 Wb = 1T×m2) B Electromagnetic Induction • Faraday’s Law: – Emf = ε = - (Δ Φm )/Δt – For a coil of N loops: ε = -N (Δ Φm )/Δt – Lenz’s Law: The current and emf induced by the changing magnetic flux is in a direction so as to oppose the change in flux. Applications 1. Motional emf : A conducting rod rides on conducting rails with velocity (v) to the right as shown, in a uniform magnetic field as shown. Given B, L, and v, what is the emf generated? Motional Emf Applet Applications (cont’d) 2. Electric Generators: How power companies generate electricity for cities and homes. • • • Relative motion of magnet and coil is what produces emf. Moving a magnet in and out of a coil produces an alternating current. Rotating a loop (or coil) between the poles of a magnet causes AC. Electric Generators (cont’d) • Current is induced in lower and upper sides of the wire loop. •Upper side has v coming out of board, resulting in a current to the right. •Lower side has v going into the board, resulting in a current to the left. •Currents add to cause clockwise current. • Electric Generators (cont’d) • Angle of Area Vector and B-field fluctuates between θ = 0° and 180 ° • Current and emf reverses direction with angular frequency (ω): θ = ωt • ε = - (Δ Φm )/Δt = -BA Δ(cos (ωt))/ Δt = BA ω sin (ωt) (using calculus) You are not responsible for this formula; just the concept of how AC is produced. Result: Alternating Current (AC) Power Generation • Fuel (Nuclear, coal, water, wind) is used to create steam • Steam drives a turbine. • The turbine rotates a coil of wires between a the poles of a strong magnet. • Alternating Current is produced • Faster Rotation causes larger current. • AC produced is carried to cities and homes. The Big Picture of Electrical Power Creation The Big Picture of Electrical Power Delivery Transformers (Devices with Two Coils): Used to adjust Voltage/Current Characteristics Power is conserved: P = VHIGHILOW = VLOWIHIGH Power is Transmitted more efficiently at: VHIGHILOW E&M • A changing E-field (or voltage) causes a changing B-field. – We saw this with an electromagnet. • A changing B-field causes a changing E-field (or voltage) – We saw this by moving a magnet in a coil • Maxwell’s Equations: – Four equations summarizing Classical Electrodynamics • Electromagnetic Waves: – Transfer Electric and Magnetic energy (Radiation) – Have oscillating E-fields and B-fields. Maxwell’s Equations: E&M Summarized