Induced Voltage and Magnetic Flux (E13) - Notes

Induktionsgesetz (E13) - Induced Voltage and Magnetic Flux

To verify that a time-varying magnetic flux, caused by a change in flux density, induces a voltage.
To differentiate between two key effects: self-induction and mutual induction.

Definition: For a conductor loop enclosing an area A with the surface normal \vec{n} , penetrated by the magnetic flux density \vec{B} , the enclosed flux \Phi is defined as:
\Phi = \int_A \vec{B} \cdot \vec{n} dA (1)

States that the temporal change of \Phi induces a voltage U{ind} in the conductor loop: U{ind} = -\dot{\Phi} (2)
Lenz's Rule: The negative sign in equation (2) corresponds to Lenz's rule. This rule dictates that the induced voltage (and thus the induced current and its accompanying magnetic field) attempts to counteract the change in flux that caused it.
Sources of Flux Change: \dot{\Phi} can be generated either by a temporal change of the magnetic field \vec{B} or by a change in the area A penetrated by \vec{B} .

For a long, air-filled cylindrical coil, the magnitude of the magnetic flux density \vec{B} is given by: |\vec{B}| = N1 \frac{I}{l} \mu0 (3)
- N_1 : number of turns of the coil.
- l : length of the coil.
- I : current flowing through the coil.
- \mu_0 : permeability of free space.
Application with a Short Coil: Instead of a single conductor loop, a very short coil with N2 turns and a cross-sectional area A2 is used. (Quantities related to the long coil are indexed with 1, those of the short coil with 2).
Since voltage is induced in each individual turn according to Eq. (2), the total voltage generated in the entire short coil is:
U{ind} = -N2 \dot{\Phi} (4)

Equation (4) can be transformed using equations (1) and (3) to relate to the temporal change of the current flowing through the long coil. For \vec{A2} \parallel \vec{B} , the induced voltage is: U{ind} = -N2 \dot{\Phi} = -\mu0 A2 \frac{N1 N_2}{l} \dot{I} = -M \dot{I} (5)
Mutual Inductance ( M ): In the last step, the mutual inductance M was defined, which depends on the geometric arrangement of the two coils.
Process: The entire physical process described by this interaction is called mutual induction.

The change in the magnetic field generated by the long coil not only induces a voltage in the short coil but also in the long coil itself.
This phenomenon is called self-induction.
Self-Inductance ( L ): Analogous to Eq. (5), a self-inductance L can be introduced:
U{ind} = -\mu0 A1 \frac{N1^2}{l} \dot{I} = -L \dot{I} (6)
Significance: Self-induction is particularly important when a coil is operated with AC voltage.

If a coil is connected directly to a voltage source with a harmonic time dependence U(t) = U0 \cdot e^{i\omega t} , and R is the ohmic resistance of the coil, the current I flowing through the coil is given by: U0 \cdot e^{i\omega t} = L \dot{I} + R I (7)
Solution Ansatz: By substituting the solution approach I = I0 \cdot e^{i(\omega t - \phi)} into the equation and rearranging, we get: \frac{U(t)}{I(t)} = \frac{U0}{I_0} e^{i\phi} = i\omega L + R = Z (8)
Impedance ( Z ): The complex quantity Z is called impedance and represents an effective resistance for alternating current.
Reactance: The term i\omega L is also called reactance (