Induction and Inductance

• Questions??
• Current produces magnetic field
• Can magnetic field produce current



• Observations
• move a magnet through a coil
• galvanometer needle deflects
• size of deflection related to speed of magnet
• direction of deflection
• orientation of magnet
• moving the magnet changes size of B in coil
• stretch or shrink coil in magnetic field
• galvanometer needle deflects
• size of deflection related to rate of change
• direction of deflection
• orientation of magnetic field
• whether loop grows or shrinks
• stretching or shrinking changes area
• rotate coil in magnetic field
• galvanometer needle deflects
• size of deflection related to rate of rotation
• direction of deflection
• orientation of magnet
• rotating the magnet changes the angle between the coil and B
• Magnetic Flux
• Number of field lines passing through loop
• size of magnetic field
• area of loop
• angle loop makes with field



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• B = magnetic field (uniform)
• A = area of loop
• q = angle between B and the normal to the loop



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• general expression if the field is not uniform.
• magnetic flux has units of [T-m²] = [Weber]
• Faraday's Law
• Induced emf
• depends on rate of change of magnetic flux



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• one loop



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• N loops



• Use Faraday's Law to get magnitude of the induced voltage
• Lenz's Law
• The direction of the induced emf is such that it will oppose the change that created it.
• Lenz's Law will give the direction or polarity of the emf.



• Examples



• Motional Emf
• conducting rod moving through a magnetic field
• charges can move freely in the rod.
• with the rod moving as shown through a magnetic field pointing into the page, positive charges will migrate to the top end of the rod and negative charges will migrate to the bottom end of the rod.
• this will continue until the electric force balances the magnetic force.



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• if we put this rod on conducting rails and connect a load to complete the circuit
• then we can use Faraday's Law to determine the induced emf for the loop.




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• Lenz's Law
• flux is increasing downward, induced current must produce magnetic field pointing out of page
• current will be in CCW matching the polarity found using the motional emf approach.
• Examples



• Inductors
consider a very simple circuit consisting of a battery, resistor and a switch.
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• When the circuit is closed, current will flow. This produces a changing magnetic flux inside the loop of the circuit. This will induce an emf directed such that it opposes the change in current.



• With an inductor added to the circuit, we have the circuit shown. Applying the loop rule gives






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• The solution to this equation is





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• The inductor will also play a role when the switch is opened and the current is decreasing.



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• Energy Stored in a Magnetic Field
• applying the loop rule again and multiplying the equation by I



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• IV represents the rate at which energy is being provided to the circuit by the battery.
• not all of that energy is being dissipated in the resistor, some of it is going to the inductor.
• the inductor is storing energy similar to the capacitor in an RC circuit.
• when the switch is opened, the inductor will induce an emf that will try to keep current moving. After the switch is opened, the resistor will dissipate the energy stored in the inductor.
• How much energy is stored?



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