Magnetic Induction - magnetic flux - induced emf - Faraday’s Law and Lenz’s Law

Magnetic Flux Flux through a surface S: (dA is the “area vector”, perpendicular to the surface.) a scalar ; units, 1 T·m2 = 1 Weber (Wb) represents “number of magnetic field lines through a surface S”

Magnetic Flux Flux through a closed surface S: This is like Gauss’ law, but for magnetism. However, it states that: the number of magnetic field lines that enter a volume enclosed by a surface S must equal the number that leave the volume and it implies that magnetic monopoles do not exist

Faraday’s Law: When the external magnetic flux ΦB through a closed conducting loop with one turn changes, the emf induced in the closed loop is: (for a loop with N turns: ) Note that ΦB changes if: 1) B changes 2) the area of the circuit changes (dA) 3) the orientation of the circuit changes (B•dA)

Lenz’s Law (the negative sign) (for the direction of the induced emf) The induced emf and induced current direction in the loop is such that the magnetic flux it produces inside the loop opposes the change in flux inside the loop produced by the external field. The induced emf is directly proportional to the rate of change of the magnetic flux thought the circuit.

The “-” sign: Field vector B positive current direction For given B direction the R.H. rule defines a corresponding “positive” current and emf direction. e.g. B , increasing: induced ε is –ve B , decreasing: induced ε is +ve

Induction Move a magnet at constant speed through a coil attached to a voltmeter: B v S N S N +ve current direction voltmeter reading position x

Induced current creates a field in the same direction (inside the loop). ε Bexternal

Induced current creates a field in opposite direction. Bexternal ε

Circular coil, 100 turns, area = πr2 = 0.10m2 Example 1 Circular coil, 100 turns, area = πr2 = 0.10m2 x x x x B is an external magnetic field, and is changing with time as in the graph below. 0 1 1.5 2 3 4 2T t (sec.) B Plot emf, paying attention to its sign. Note: CW is the direction of positive emf

Solution 0 1 1.5 2 3 4 2T t (sec.) B ε 0 1 1.5 2 3 4 t (sec.)

Quiz A circuit of area A is made from a single loop of wire connected to a resistor of resistance R. It is placed in a uniform external field B (at right angles to the plane of the loop). B is reduced uniformly to zero in time Dt. The total charge which flows through the resistor is: independent of Dt proportional to Dt inversely proportional to Dt zero

Motional emf emf induced in a conductor moving through a magnetic field.

Conductor moving in uniform B : Force on charge q: x x x x As the positive charge moves slowly along the conductor, parallel to Fm , the work done on each charge is: B W = Fm l = qBvl The emf (work per unit charge) induced between the bar ends is: E ε = W/q Fm + v Which end of the rod is positive after some time? What is the “E ” for?

Area enclosed by closed loop = A = l x x x x x x x x We can derive the same expression, ε = Bvl, from Faraday’s Law, if we look at a simple complete circuit. Consider a conducting bar sliding along a a U-shaped conductor as shown. v l B Area enclosed by closed loop = A = l x What is the direction of the induced current?

Example 2 B R = 5Ω l = 25cm v R B = 2T l v = 3m/s Find: emf current x x x x x x v l R B R = 5Ω l = 25cm B = 2T v = 3m/s Find: emf current force to keep bar moving power to keep bar moving

Solution

Example 3 A bar of mass m and length l moves on two frictionless parallel rails in the presence of a uniform B directed into the paper. The bar is given an initial velocity vo to the right and is released. Find the: a) velocity of the bar as a function of time b) induced current c) induced emf B x x x x x x x x x vo R l

Solution

Summary Faraday’s Law: Lenz’s Law: (for direction of ε) A changing magnetic flux induces an emf in a circuit: Lenz’s Law: (for direction of ε) The induced emf causes an induced current whose flux would oppose the change in external flux through the loop.