Study Notes on Faraday's Law, Induction, and Conduction in MRI

Overview of Faraday's Law and Induced EMF

  • Understanding of Faraday's Law

    • Faraday's law states that the EMF (electromotive force) induced in a conducting loop is proportional to the rate of change of magnetic flux through the loop.

    • EMF can also be referred to as voltage, measured in volts.

    • Represented in formulas with the Greek letter epsilon (ε).

    • The fundamental relationship is given by:
      ext{EMF} = - rac{ ext{d} ext{flux}}{ ext{d}t}

    • Here, the negative sign is a consequence of Lenz's law, which states that the direction of induced EMF and hence current will oppose the change in flux.

Magnetic Flux and its Units

  • Magnetic flux ( Φ) is represented as: ext{flux} = B imes A imes ext{cos}( heta)

    • Where

      • B = magnetic field strength (measured in Teslas)

      • A = area of the loop (measured in m²)

      • θ = angle between the magnetic field and the normal to the loop area

  • Units of magnetic flux = (T imes m^2 = Wb) (Webers)

Lenz's Law

  • Lenz's law states that:

    • The direction of induced current is such that it creates a magnetic field opposing the change in the magnetic flux that produced it.

  • This is represented mathematically and conceptually to ensure conservation of energy.

Induced EMF in a Coil with n Turns

  • When considering a coil of wire with n turns, the induced EMF formula alters:
    ext{EMF} = -n imes rac{ ext{d} ext{flux}}{ ext{d}t}

  • Thus, the induced EMF is multiplied by the number of loops in the coil, allowing for increased voltage output.

Using Ohm's Law

  • Ohm's law provides a relation between voltage, current, and resistance: ext{EMF} = I imes R

    • Where

      • I = current (in Amperes)

      • R = resistance (in Ohms)

  • Thus, current can be determined from:
    I = rac{ ext{EMF}}{R}

Practical Implications: MRI Example

  • A real-life scenario involving induced EMF:

    • Patient undergoing MRI left on a copper bracelet:

    • A copper bracelet can behave as a loop of wire in a changing magnetic field during an MRI scan.

    • As magnetic field strength changes, current can be induced in the bracelet leading to potential heating and burns.

    • Example specifics:

    • Diameter of the bracelet = 6 cm (radius = 3 cm).