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Physics 272 March 17 Spring 2015 www.phys.hawaii.edu/~philipvd/pvd_15_spring_272_uhm go.hawaii.edu/KO Prof. Philip von Doetinchem [email protected] PHYS272 - Spring 15 - von Doetinchem - 133 Direction of induced electromagnetic fields ● For increasing external magnetic field the induced magnetic field is in the opposite direction and works against the external field ● For decreasing external magnetic field the induced magnetic field is in the same direction ● Sign rules for the direction of induced emf: – Define positive direction of area – Determine the sign of the magnetic flux from the area and the magnetic field – If flux is increasing → induced emf is negative If flux is decreasing → induced emf is positive – Right hand rule: ● ● ● align area vector with thumb Positive emf → current is in the same direction as curled fingers Negative emf → current is in the opposite direction of curled fingers Not magnetic flux, but changing magnetic flux causes induction effects PHYS272 - Spring 15 - von Doetinchem - 134 Lenz's law ● ● ● ● ● Alternative method for determining the direction of induced current or emf Lenz's law can be derived from Faraday's law The direction of any magnetic induction effect is such as to oppose the cause of the effect Cause can be – Changing flux due to varying magnetic field – Changing flux due to motion of conductors Source: http://de.wikipedia.org/wiki/Emil_Lenz Heinrich F. E. Lenz 1804-1865 Think about it like: induced current tries keeping the system in the state it was before the flux change happened. PHYS272 - Spring 15 - von Doetinchem - 135 Slidewire generator ● ● Look at individual charge in slidewire: – Feels magnetic force – Separates charges – Builds up electric field – Equilibrium between electric force and magnetic force (→ also see Hall effect) No magnetic forces act on the charges in the stationary U part, but sliding rod creates potential difference (source of emf) → establishes current PHYS272 - Spring 15 - von Doetinchem - 139 Motional electromotive force ● ● ● Origin of electromotive force is of non-electrostatic nature (similar to battery → chemical) Charges are brought to a higher potential Concept can be generalized to conductors of any shape and in any field (can be non-uniform, but not varying with time) – take the perpendicular projection of the velocity with respect to the magnetic field (cross product) – Use the parallel projection of the former along a line element of the conductor (scalar product) PHYS272 - Spring 15 - von Doetinchem - 140 Induced electric fields ● ● We understand the concept of induction for moving charges Where is induced current coming from if the flux is changing in a stationary conductor? PHYS272 - Spring 15 - von Doetinchem - 143 Induced electric fields ● ● Green wire loop is not in a magnetic field (magnetic field outside solenoid is negligible) Only the magnetic flux through the loop is changing PHYS272 - Spring 15 - von Doetinchem - 144 Induced electric fields ● ● ● Before: charges were pushed through conductor because of magnetic forces Conclusion for stationary case: changing magnetic flux generates an induced electric field in the wire loop Furthermore: – induced electric field in the loop is not conservative – charges gain electric potential Stationary setup PHYS272 - Spring 15 - von Doetinchem - 145 Induced electric fields ● What does the electric field look like? ● Cylindrical symmetry → electric field has the same magnitude on the circle → has to be tangential to cancel out according to Gauss's law (no net charge present inside) ● Line integral has to be negative when magnetic flux is increasing (Lenz's law) PHYS272 - Spring 15 - von Doetinchem - 146 Induced electric fields ● ● ● The work done on an electron by the induced electric field during a complete trip around the loop is e ε energy can be removed from the electron due to the resistance of the loop The induced electric field is a non-conservative field → path does matter in this case, not just the potential difference PHYS272 - Spring 15 - von Doetinchem - 147 Nonelectrostatic electric fields ● ● ● Faraday's law works for two different situations: – Induced current from magnetic forces when conductor moves through magnetic field – Time-varying magnetic field induces electric field in a stationary conductor and induces a current The electric field of the 2nd case is also induced when no conductor is present – It is not conservative – Field does non-zero amount of work on charges particle on closed path – This is a non-electrostatic electric field in contrast to a electrostatic electric field A change of magnetic field acts as a source of electric field that cannot be produced with a static distribution PHYS272 - Spring 15 - von Doetinchem - 148 Displacement current and Maxwell's equations ● ● ● ● A varying magnetic field creates an induced electric field Varying electric fields also create magnetic fields Essential feature to understand electromagnetic waves To understand relationship: look at charging of capacitor PHYS272 - Spring 15 - von Doetinchem - 152 Displacement current and Maxwell's equations ● Look at charging of capacitor: – Conducting current ic charges capacitor and builds up electric field – No conducting current between plates – Applying Ampere's law to both situations reveals contradiction: PHYS272 - Spring 15 - von Doetinchem - 153 Displacement current and Maxwell's equations ● Electric flux increases while conducting current is decreasing ● Charge on capacitor: ● Charging capacitor → current changes: PHYS272 - Spring 15 - von Doetinchem - 154 Displacement current and Maxwell's equations ● Discrepancy from last slides can be resolved by having the change in conducting current translate into a change of electric flux ● Ampere's law becomes: ● Displacement current density: ● In this sense the displacement current is going through the capacitor PHYS272 - Spring 15 - von Doetinchem - 155 The reality of displacement current ● ● Physical significance of displacement current? This magnetic field can me measured and has a real physical meaning PHYS272 - Spring 15 - von Doetinchem - 156 Maxwell's equations of electromagnetism ● ● Gauss's law for electric fields (surface integral) – Electric field is related to total charge in an enclosed surface – Electric charges are sources of magnetic fields Gauss's law for magnetism (surface integral) – No magnetic monopoles exist → magnetic flux through closed surface is always zero PHYS272 - Spring 15 - von Doetinchem - 157 Maxwell's equations of electromagnetism ● Ampere's law (line integral) – ● Conducting and displacement current act as sources of magnetic fields Faraday's law (line integral) – A changing magnetic field or magnetic flux induces an electric field PHYS272 - Spring 15 - von Doetinchem - 158 Maxwell's equations of electromagnetism ● ● ● Electric field in Maxwell's equation is a superposition of – the conservative part from the electrostatic field caused by a charge distribution (does not contribute to line integral in Faraday's law) – The non-conservative part caused by induced currents (does not contribute to surface integral in Gauss's law as it is not caused by static charges) Time-varying field of either kind induce field of the other kind Starting point for electromagnetic wave discussion → physical basis for light, X-ray, etc. PHYS272 - Spring 15 - von Doetinchem - 159 Additional material PHYS272 - Spring 15 - von Doetinchem - 160 Induced electric fields ● solenoid with 500 turns, A=4.0cm2, current in windings is increasing with 100A/s PHYS272 - Spring 15 - von Doetinchem - 161 Motional emf in the slidewire generator ● Suppose the moving rod in the slidewire configuration is 0.1m long and 2.5m/s fast with a resistance of 0.03 and a uniform magnetic field of 0.6T: – Induced current in the loop: – Force on the rod PHYS272 - Spring 15 - von Doetinchem - 162 The Faraday disk dynamo ● As before for the slidewire generator: assume positive free charge carriers ● Positive charges accumulate at the edges ● Radially outward current flow ● Electric field builds up → emf is created ● Difference: velocity depends on the distance to the center PHYS272 - Spring 15 - von Doetinchem - 163 Bar on inclined plane PHYS272 - Spring 15 - von Doetinchem - 164 Bar on inclined plane PHYS272 - Spring 15 - von Doetinchem - 165 Bar on inclined plane PHYS272 - Spring 15 - von Doetinchem - 166 Bar on inclined plane PHYS272 - Spring 15 - von Doetinchem - 167 Eddy currents ● ● ● Induced currents are not necessarily confined to well-defined paths in conductors Induced eddy-like currents can form in any type of metal in changing magnetic fields or by moving through a magnetic field Applications: – Currents causes heating → induction furnace – Eddy currents causes braking effect → trains – Metal detector at the airport: ● ● ● Magnetic field creates eddy current in objects Eddy current creates induced magnetic field Induced magnetic field creates eddy currents in receiver coil PHYS272 - Spring 15 - von Doetinchem - 168 Direction of eddy currents upper current: falls through region of increased magnetic field → builds up induced magnetic field against external field (counter-clockwise current) metallic disk falling through magnetic field - currents to the right are induced - induced currents feel upward magnetic force → slow down velocity stationary magnetic field only disk is moving magnetic field is stationary lower current: falls through region of decreased magnetic field → builds up induced magnetic field trying to maintain the external field (clockwise current) PHYS272 - Spring 15 - von Doetinchem - 169 Eddy currents https://www.youtube.com/watch?v=7_-RqkYatWI ● ● solid copper pendulum mounted between poles of an electromagnet – pendulum is set into motion – then the magnets are turned on – magnets induce eddy currents in the copper opposing the motion of the pendulum – pendulum quickly slows to a stop – eddy current braking copper pendulum with strips cut into it is not slowed nearly as much as the solid pendulum – cuts in the copper prevent large eddy currents from forming – only eddy currents smaller than strips of copper can be formed PHYS272 - Spring 15 - von Doetinchem - 170 Eddy currents https://www.youtube.com/watch?v=Pl7KyVIJ1iE ● ● solid metal ring placed on iron core whose base is wrapped in wire – when DC current is passed through the wire, a magnetic field is formed in the iron core – this sudden magnetic field induces a current in the metal ring, which in turn creates another magnetic field that opposes the original field – ring briefly jumps upwards cut in the ring – ● cannot form current inside → will not jump ring is cooled in liquid nitrogen → resistance of the metal is lowered → more current to flow. – ring jump jumps higher ● magnetic field curves away at the top of the iron coil → with DC power ring will never fly off the top ● When AC current is passed through wire → ring flies off the top of the iron core. – current lags the emf by 90 degrees in inductors – forces on the ring are always pointing upwards PHYS272 - Spring 15 - von Doetinchem - 171 Conducting and displacement current ● Rod of pure silicon is carrying a current. Electric field varies sinusoidal with time. PHYS272 - Spring 15 - von Doetinchem - 172 Conducting and displacement current ● Rod of pure silicon is carrying a current. Electric field varies sinusoidal with time. PHYS272 - Spring 15 - von Doetinchem - 173