This study note covers the essential topics of Induction, Alternating Currents, and Capacitance as specified in the International Baccalaureate (IB) Physics syllabus. The focus will be on understanding the fundamental principles, mathematical formulations, and practical applications. The document is structured to break down complex ideas into digestible sections with examples, tips, and notes to aid comprehension.
Electromagnetic induction occurs when a conductor moves through a magnetic field, creating a potential difference along the wire. This phenomenon can be observed when a magnet is moved near a coil of wire. The movement of the magnet induces an electromotive force (emf) in the wire, causing a current to flow if the circuit is closed.
Key Points:
Magnetic flux ($\Phi$) is defined as the product of the magnetic flux density ($B$) and the cross-sectional area ($A$) perpendicular to the direction of the magnetic flux density.
$$ \Phi = B \cdot A $$
Where:
Changing Angle:
Note:
The amount of magnetic flux varies as the coil rotates within the field.
Induced emf is generated when a conductor cuts through magnetic field lines, causing a change in magnetic flux ($\Delta \Phi$). This change results in work being done, which is transformed into electrical energy.
Mathematical Formulation: $$ \varepsilon = - \frac{d(N \Phi)}{dt} $$
Where:
Example:
Example Calculation: A small rectangular coil with 350 turns of wire, longer sides of 3.5 cm, and shorter sides of 1.4 cm is rotated in a magnetic field of flux density 80 mT. The coil is turned through an angle of 40° in 0.18 seconds. Calculate the induced emf.
Solution:
Using Faraday's Law: $$ \varepsilon = \frac{N \Delta \Phi}{\Delta t} $$ $$ \varepsilon = \frac{350 \times (80 \times 10^{-3} \times 4.9 \times 10^{-4})}{0.18} $$ $$ \varepsilon \approx 0.076 V $$
Faraday’s Law states that the induced emf in a coil is directly proportional to the rate of change of magnetic flux linkage.
Lenz's Law states that the direction of the induced current (and hence emf) is such that it opposes the change in magnetic flux that produced it. This is represented by the negative sign in Faraday’s Law.
Note:
Lenz's Law is a direct consequence of the principle of conservation of energy.
An AC generator converts mechanical energy into electrical energy in the form of alternating current (AC).
Example:
Example of AC Generator: A simple alternator consists of a rotating coil in a magnetic field. As the coil rotates, it cuts through the magnetic field lines, inducing an alternating emf and current.
The root-mean-square (RMS) values of current and voltage are used to express the effective values of AC.
$$ I_{rms} = \frac{I_{peak}}{\sqrt{2}} $$ $$ V_{rms} = \frac{V_{peak}}{\sqrt{2}} $$
Where:
Transformers are used to change the voltage levels in AC circuits.
For an ideal transformer: $$ \frac{V_s}{V_p} = \frac{N_s}{N_p} $$ $$ \frac{I_s}{I_p} = \frac{N_p}{N_s} $$
Where:
Tip:
Remember that power in an ideal transformer is conserved: $P_p = P_s$.
Capacitance ($C$) is the ability of a system to store electric charge per unit voltage.
$$ C = \frac{Q}{V} $$
Where:
Dielectrics are insulating materials placed between the plates of a capacitor to increase its capacitance by reducing the electric field.
The time constant ($\tau$) of an RC circuit is the time taken for the voltage to either charge or discharge to approximately 63% of its final value.
$$ \tau = R \cdot C $$
Where:
The energy ($E$) stored in a capacitor is given by:
$$ E = \frac{1}{2} C V^2 $$
Example:
Example Calculation: A capacitor with capacitance 10 µF is charged to a voltage of 5V. Calculate the energy stored.
Solution: $$ E = \frac{1}{2} \times 10 \times 10^{-6} \times 5^2 $$ $$ E = 0.125 \times 10^{-3} \text{ J} $$ $$ E = 0.125 \text{ mJ} $$
The voltage across a capacitor during charging and discharging can be described by exponential functions.
Where:
Common Mistake:
A common mistake is to forget that the time constant $\tau$ depends on both resistance and capacitance.
To calculate the time required for a capacitor to discharge to a certain voltage:
$$ t = -\tau \ln\left(\frac{V}{V_0}\right) $$
Where:
By understanding these fundamental concepts and equations, students can effectively tackle problems related to induction, alternating currents, and capacitance in the IB Physics syllabus.