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Induction Faraday’s Law

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Induction We will start the discussion of Faraday’s law with the description of an experiment. A conducting loop is connected to a sensitive ammeter. Since there is no battery in the circuit there is no current.

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Induction

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Whenever there is a change in the number of field lines passing through a loop of wire a voltage (emf) is induced (generated). More formally: The magnitude of the emf induced in a conducting loop is equal to the rate at which the magnetic flux through the loop changes with time.

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Induction Faraday’s law expresses this phenomena, Where the magnetic flux through the loop is given by the closed integral,

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Induction For a coil with N turns Faraday’s law becomes,

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Induction In general the induced emf tends to oppose the change in flux producing it. This opposition is indicated by the negative sign in equation for Faraday’s law.

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Induction The general means of changing the flux are; 1.Change the magnitude of the magnetic field within the coil. 2.Change the area of the coil or the area cutting the field. 3.Change the angle between B and dA.

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Induction Lenz’s Law

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Induction The direction of the induced current in a loop is determined from Lenz’s law. The law states that: An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux that induces the current.

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Consider. The direction of increasing B is to the left. The direction opposing this is to the right. Using the screw rule point the thumb in the direction opposing the change. The fingers give the direction of the induced current.

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Induction

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Let us look at the following case as an example of induction. Let us look at what happens as a conduction moves through a magnetic field. There is a change in the area of flux cut hence an induced i and an emf. Remember where

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Induction Consider a conductor (length L) sliding along a rail with a velocity v in an uniform magnetic field B. xxxx xxxx xxxx v

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Induction As the conductor moves through the field electrons are push upward (Fleming’s left hand rule) making the top –ve and the bottom +ve. xxxx xxxx xxxx v -ve +ve x

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Fleming’s left hand: 1 st finger: magnetic field, 2 nd current and the thumb direction of movement. We can also use the Lorentz law.

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Induction This induces an electrostatic field and emf across the conductor (induced emf) which acts as a source. The direction of the induced current (conventional current) is clockwise. xxxx xxxx xxxx v -ve +ve x I

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Induction The flux cut as the conductor moves through the field is, xxxx xxxx xxxx v -ve +ve x I

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Induction The flux cut as the conductor moves through the field is, Rate of change of flux, xxxx xxxx xxxx v -ve +ve x I

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Induction Therefore, Hence from Faraday’s Law the induced emf is, xxxx xxxx xxxx v -ve +ve x I

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Ampere-Maxwell Law

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Recall Ampere’s Law,

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Ampere-Maxwell Law Recall Ampere’s Law, Ampere’s Law can be modified as follows to incorporate the findings of Maxwell,

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Ampere-Maxwell Law Recall Ampere’s Law, Ampere’s Law can be modified as follows to incorporate the findings of Maxwell, That is, there are two ways for a magnetic field to be formed:

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Ampere-Maxwell Law 1.By a current (given by Ampere’s Law ). 2.By a change in flux ( ).

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Ampere-Maxwell Law 1.By a current (given by Ampere’s Law ). 2.By a change in flux ( ). The later part of the equation governing the induction of a magnetic field.

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Ampere-Maxwell Law That is, a magnetic field is induced along a closed loop by changing the electric flux in the region encircled by the loop.

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Ampere-Maxwell Law That is, a magnetic field is induced along a closed loop by changing the electric flux in the region encircled by the loop. An example of this induction occurs during the charging of a parallel plate capacitor.

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Ampere-Maxwell Law Ex: Consider a parallel-plate capacitor with circular plates of radius R which is being charged. Derive an expression for B at radii r for r ≤ R and r ≥ R.

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Ampere-Maxwell Law Using the methodology of Ampere’s Law we draw a closed loop between the plates.

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Ampere-Maxwell Law Recall, There is no current between the plates,

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Ampere-Maxwell Law Recall, There is no current between the plates, For

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Ampere-Maxwell Law Note:

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Ampere-Maxwell Law The equation tells us that B increases linearly with increasing radial distance r.

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Ampere-Maxwell Law For In this case,

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Ampere-Maxwell Law B decreases as1/r.

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Ampere-Maxwell Law linear decay 1/r Rr B

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Ampere-Maxwell Law Comparing the the two terms on the right of the Ampere-Maxwell equation, we see that that the two terms must have dimensions of current. ie. The dimensions of and must be the same

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Ampere-Maxwell Law The later product will be treated as a current and is called the displacement current.

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Ampere-Maxwell Law The later product will be treated as a current and is called the displacement current. Therefore Ampere-Maxwell’s Law can be rewritten as,

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Ampere-Maxwell Law The direction of the magnetic field is found by assuming the direction of the displacement current is that of the current. Then use the screw rule.

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Ampere-Maxwell Law idid

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Rewriting the results of the circular capacitor using the displacement current,

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Ampere-Maxwell Law Remember the displacement current is not a flow of electrons.

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Faraday’s Law Induced Electric fields

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From the previous discussion of Faraday’s law we recognise that, : a conducting ring placed in magnetic field of changing strength will induce an emf which in turn will induce a current.

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The induced emf and current are illustrated to the right.

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Since a magnetic field can’t directly produce a current it must be due to an electric field.

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What we find is that the changing magnetic flux through the ring produces an electric field. The electric field provides the work need to move charge around the ring.

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The work done in moving a charge q 0 around a ring is:

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So that,

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Therefore Faraday’s Law can be reformulated as,

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The electric field exists independent of the conductor! Permeates all of the space within the region of changing magnetic field. The red lines indicate the electric field lines.

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Consider a ring of radius 8.5cm in a magnetic field which changes as 0.13T/s. Find the expression for the induced electric field at a radius of 5.2cm from the centre.

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Recall: LHS: RHS: Thus:

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THE END

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