Lecture 27 Chapter 32 Magnetism of Matter Review 3 types of magnetism Diamagnetism Paramagnetism Ferromagnetism Explain how materials exhibit different types of magnetism using electron s spin and orbital magnetic dipole moments Placing material in an external B field causes dipole moments to align creating an induced B field Degree of alignment determines type of magnetism and amount of magnetization Review Magnetic flux through an area A in a B field is r r B B dA Induced emf occurs when magnetic flux changes with time d B E N dt Changing B field induces an E field Faraday s law r r d B E d s dt Magnetism 18 Magnetic monopoles do not exist Express mathematically as r r B B dA 0 Integral is taken over surface Net magnetic flux through closed surface is zero closed As many B field lines enter leave the surface as Magnetism 19 Gauss s law for E fields r r qenc E E dA 0 Gauss s law for B fields r r B B dA 0 Both cases integrate over closed Gaussian surface Magnetism 20 Faraday s law of induction E field is induced along a closed loop by a changing magnetic flux encircled by that loop Is the reverse true Maxwell s law of induction B field is induced along a closed loop by a changing electric flux in region encircled by loop r r d B E ds dt r r d E B ds 0 0 dt Magnetism 21 Consider circular parallelplate capacitor with E field increasing at a steady rate While E field changing B fields are induced between plates both inside and outside point 1 and 2 If E field stops changing B field disappears r r d E B ds 0 0 dt Magnetism 22 r r d B E d s dt r r d E B ds 0 0 dt Two differences Extra symbols 0 and 0 to preserve SI units Minus sign means induced E field and induced B field have opposite directions when produced in similar situations Magnetism 23 Ampere s law r r B ds 0ienc Combine Ampere s and Maxwell s law r r d E B ds 0 0 dt 0ienc B field can be produced by a current and or a changing E field Wire carrying constant current d E dt 0 Charging a capacitor no current so ienc 0 Magnetism 24 What is the induced B field inside a circular capacitor which is being charged r r d E B ds 0 0 dt 0ienc No current between capacitor plates so ienc 0 and equation becomes r r d E B ds 0 0 dt Magnetism 25 For left hand side of equation chose Amperian loop inside capacitor r r B ds Bds cos B and ds are parallel and B is constant so r r B ds Bds cos 0 B ds B 2 r Magnetism 26 For right hand side of equation find E flux Amperian loop E uniform between plates to area A of loop through and r r E E dA EA Right hand side of equation becomes d E d dE 0 0 0 0 EA 0 0 A dt dt dt Magnetism 27 Equating two sides gives dE B 2 r 0 0 A dt A is area of loop A r 2 Solving for B field inside B increases linearly capacitor gives with radius 0 0 r dE B 0 at center and B max at plate edges 2 dt Magnetism 28 What is the induced B field outside a circular capacitor which is being charged r r d E B ds 0 0 dt 0ienc Realize ienc 0 and find same relations r r B d s B 2 r d E dE 0 0 0 0 A dt dt Magnetism 29 dE B 2 r 0 0 A dt E field only exists between plates so area of E field is not full area of loop only area of plates 2 A R B field becomes B 0 0 R dE 2 2r dt Outside capacitor B decreases with radial distance from a max value at r R Magnetism 30 r r d E B ds 0 0 dt 0ienc Can represent change in electric flux with a fictitious current called the displacement current id d id 0 Ampere Maxwell s law becomes r r B ds 0id enc 0ienc dt E Magnetism 31 r r B d s i i 0 d enc 0 enc Think of displacement current as fictional current between plates Use right hand rule to find direction of B field for both currents Magnetism 32 Used Ampere s law to calculate B field inside a long straight wire with current i 0i B r 2 2 R Find B field inside a circular capacitor just replace i with displacement current id 0id B r 2 2 R r r B ds 0ienc Magnetism 33 Used Ampere s law to calculate B field outside a long straight wire with current i 0i B 2 r Find B field outside a circular capacitor just replace i with displacement current id 0id B 2 r r r B ds 0ienc Magnetism 34 Checkpoint 6 Parallel plate capacitor of shape shown Dashed lines are paths of integration Rank the paths according to the magnitude of integral Bds when capacitor is discharging greatest first r r B ds 0id enc 0ienc Only displaced current in r r B ds 0id enc capacitor What is id for each path b c d all tie then a Magnetism 35 Basis of all electrical and magnetic phenomena can be described by 4 equations called Maxwell s equations As fundamental to electromagnetism as Newton s law are to mechanics Einstein showed that Maxwell s equations work with special relativity Maxwell s equations basis for most equations studied since beginning of semester and will be basis for most of what we do the rest of the semester Magnetism 36 Maxwell s 4 equations are Gauss Law r q enc r E dA 0 Gauss Law for magnetism r r B dA 0 r r d B Faraday s Law E ds dt r r d E Ampere Maxwell Law B ds 0 0 0ienc dt
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