Chapter 30: Induction and Inductance - Detailed Notes

Currents Create Magnetic Fields

  • The magnetic field (B) due to a long straight wire carrying a current i is given by: B = \frac{\mu0 i}{2\pi r}, where \mu0 is the permeability of free space and r is the distance from the wire.
  • The magnetic field (B) due to a complete loop carrying a current i is given by: B = \frac{\mu_0 i}{2R}, where R is the radius of the loop.
  • The magnetic field (B) inside a solenoid is given by: B = \mu_0 i n, where n is the number of turns per unit length.
  • The magnetic field (B) inside a torus carrying a current i is given by: B = \frac{\mu_0 i N}{2\pi r}, where N is the total number of turns and r is the radius of the torus.

Faraday’s Law of Induction

  • Magnetic Flux: If a loop enclosing an area A is placed in a magnetic field B, the magnetic flux \PhiB through the loop is defined as \PhiB = \int B \cdot dA.
  • For a uniform magnetic field B perpendicular to the plane of the loop, the magnetic flux simplifies to \Phi_B = BA.
  • The SI unit for magnetic flux is the tesla-square meter, also known as the weber (Wb): 1 Wb = 1 T⋅m².
  • If we change the magnetic flux through a coil of N turns, an induced emf appears in every turn, and the total induced emf in the coil is the sum of these individual induced emfs.
  • If the coil is tightly wound, so that the same magnetic flux \PhiB passes through all the turns, the total emf induced in the coil is: \E = -N \frac{d\PhiB}{dt} (coil of N turns).
  • Ways to change magnetic flux through a coil:
    1. Change the magnitude B of the magnetic field within the coil.
    2. Change either the total area A of the coil or the portion of that area that lies within the magnetic field.
    3. Change the angle between the direction of the magnetic field B and the plane of the coil.

Electromagnetic Induction

  • An electric current produces a magnetic field. The question is whether magnetic fields can produce electric currents.
  • Faraday's experiment: When a battery is connected to wire 1 (primary coil), current flows, creating a magnetic field. The iron ring magnifies this magnetic field. Wire 2 (secondary coil) should feel this magnetic field.
  • If a magnetic field generates an electric current, wire 2 will carry a current, which can be detected by a galvanometer.
  • Faraday observed a brief deflection of a galvanometer when the current in the primary coil was first started or when it was interrupted.
  • The galvanometer deflected one way when the primary was first connected to the battery and the opposite direction when the contact was broken.
  • No current was detected in the secondary coil when there was a steady current in the primary coil.
  • An electric current is only induced in the secondary coil when there is a changing current in the primary. This process is known as electromagnetic induction.
  • The changing current in the primary coil implies a changing magnetic field.
  • The electric current in the secondary coil implies that there is an electric field being induced.

Magnetic Flux Calculation

  • Magnetic flux is defined as: \Phi_B = \int B \cdot dA for a non-uniform magnetic field B.
  • Magnetic flux is a scalar quantity.
  • In a uniform magnetic field, the magnetic flux can be expressed as \Phi_B = BA \cos{\theta}, where \theta is the angle between the magnetic field and the normal to the loop.
  • The SI unit for magnetic flux is the weber (Wb): 1 Wb = 1 T⋅m².
  • Magnetic flux is at a maximum (\Phi = BA) when the field lines are perpendicular to the plane of the loop and zero when the field lines are parallel to the plane of the loop.

Induced EMF

  • The magnitude of the emf (electromotive force) induced in a conducting loop is equal to the rate at which the magnetic flux through that loop changes with time: \E = -\frac{d\Phi_B}{dt}.
  • If a coil consists of N loops with the same area, the total induced emf in the coil is given by: \E = -N \frac{d\Phi_B}{dt}.
  • In a uniform magnetic field, the induced emf can be expressed as: \E = -\frac{d}{dt}(BA \cos{\theta}).
  • Emf can be induced in several ways:
    • The magnitude of B can change with time.
    • The area enclosed by the loop can change with time.
    • The angle between B and the normal to the loop can change with time.
    • Any combination of the above can occur.

Example Calculation

A square loop of copper wire is placed into a magnetic field of 0.5 T. The length of each side of the wire is 12 cm.

a) Calculate the max magnetic flux that pass the loop of wire.
b) Calculate the magnetic flux when A and B are at 60°.
c) What is the magnitude of emf induced as the loop is rotated so that A and B are 60°. Assume we begin at \theta = 0 and it takes 0.2 second for rotation to take place.

Lenz’s Law

  • Lenz's Law: The direction of the current i induced in a loop is such that the current’s magnetic field B_{ind} opposes the change in the magnetic field inducing i.
  • The field is always directed opposite an increasing field and in the same direction as a decreasing field B.
  • The curled–straight right-hand rule gives the direction of the induced current based on the direction of the induced field.
  • Opposition to Pole Movement: When the north pole of a magnet approaches a closed conducting loop, the increasing magnetic flux induces a current in the loop. To oppose this increase, the loop's north pole (and magnetic moment m) faces the approaching north pole, repelling it. The induced current is counterclockwise.
  • If the magnet is pulled away, the loop forms a south pole to attract the retreating north pole, and the induced current is clockwise.

Example

A coil of wire with 50 turns has a uniform magnetic field of 0.4 T passing through the coil perpendicular to its plane. the coil encloses an area of 0.03 m². If the flux through the coil is reduced to zero by removing it from the field in a time of 0.25 s, what is the induced voltage in the coil?