induction faraday’s law. induction we will start the discussion of faraday’s law with the...
TRANSCRIPT
Induction
Faraday’s Law
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.
Induction
Induction
• 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.
Induction
• Faraday’s law expresses this phenomena,
• Where the magnetic flux through the loop is given by the closed integral,
dt
d B
AdBB
Induction
• For a coil with N turns Faraday’s law becomes,
dt
dN B
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.
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.
Induction
Lenz’s Law
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.
• 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.
Induction
Induction
• 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 wheredt
dN B
AdBB
Induction
• Consider a conductor (length L) sliding along a rail with a velocity v in an uniform magnetic field B.
x x x x
x xxx
x x x x
v
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.
x x x x
x xxx
x x x x
v
-ve
+ve
x
• Fleming’s left hand:
• 1st finger: magnetic field, 2nd current and the thumb direction of movement.
• We can also use the Lorentz law.
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.
x x x xx xxx
x x x x
v
-ve
+ve
xI
Induction
• The flux cut as the conductor moves through the field is, BAdABAdBB
.
x x x xx xxx
x x x x
v
-ve
+ve
xI
)(LxBB
Induction
• The flux cut as the conductor moves through the field is,
• Rate of change of flux,
BAdABAdBB
.
x x x xx xxx
x x x x
v
-ve
+ve
xI
)(LxBB
)(LxBdt
d
dt
d B
dt
dxBL
Induction
• Therefore,
• Hence from Faraday’s Law the induced emf is,
x x x xx xxx
x x x x
v
-ve
+ve
xI
BLvdt
d B
BLvdt
d B
Ampere-Maxwell Law
Ampere-Maxwell Law
• Recall Ampere’s Law, encisdB 0.
Ampere-Maxwell Law
• Recall Ampere’s Law,
• Ampere’s Law can be modified as follows to incorporate the findings of Maxwell,
encisdB 0.
dt
disdB Eenc
000.
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:
encisdB 0.
dt
disdB Eenc
000.
Ampere-Maxwell Law
1. By a current (given by Ampere’s Law
).
2. By a change in flux ( ).
N
nnenc ii
100
dt
d E00
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.
N
nnenc ii
100
dt
d E00
dt
dsdB E 00.
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.
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.
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.
Ampere-Maxwell Law
• Using the methodology of Ampere’s Law we draw a closed loop between the plates.
Ampere-Maxwell Law
• Recall,
• There is no current between the plates,dt
disdB Eenc
000.
dt
dsdB E 00.
Ampere-Maxwell Law
• Recall,
• There is no current between the plates,
• For
dt
disdB Eenc
000.
dt
dsdB E 00.
Rr
dt
drB E
002
Ampere-Maxwell Law
• Note: 2rEEAE
dt
rEdrB
2
002
dt
dEr 2
00
dt
dErB
200
Ampere-Maxwell Law
• The equation tells us that B increases linearly with increasing radial distance r.
Ampere-Maxwell Law
• For
• In this case,
Rr dt
dsdB E 00.
2REEAE
dt
dE
r
RB
2
200
Ampere-Maxwell Law
• B decreases as1/r.
Ampere-Maxwell Law
linear
decay 1/r
R r
B
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
dt
disdB Eenc
000.
encidt
d E0
Ampere-Maxwell Law
• The later product will be treated as a current and is called the displacement current .di
dt
di Ed
0
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,
di
dt
di Ed
0
encencd iisdB 0,0.
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.
Ampere-Maxwell Law
id
Ampere-Maxwell Law
• Rewriting the results of the circular capacitor using the displacement current,
Rr r
iB d
2
0
20
2 r
iB d
Rr
Ampere-Maxwell Law
• Remember the displacement current is not a flow of electrons.
Faraday’s Law
Induced Electric fields
• 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.
• The induced emf and current are illustrated to the right.
• Since a magnetic field can’t directly produce a current it must be due to an electric field.
• 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.
• The work done in moving a charge q0 around a ring is:
sdEqsdFW
0.
• The work done in moving a charge q0 around a ring is:
• So that, sdEqsdFW
0.
sdE
• Therefore Faraday’s Law can be reformulated as,
dt
dsdE B
• 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.
• 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.
• Recall:
• LHS:
• RHS:
• Thus:
dt
dsdE
dsEEdssdE
rE 2
2rBBAAdB
dt
dBrrE 22
dt
dBrE
2 mVmE 4.3
THE END