Topic 11: Electromagnetic Induction

11.1 Electromagnetic Induction

Magnetic flux density: B (Tesla). ‘How concentrated the field lines are’

Magnetic flux: B.A (T m2 -> Weber). ‘How many field lines cut a surface perpendicularly’

Magnetic flux linkage: nB.A

Faraday’s Law of Electromagnetic Induction: The electromotive force induced is directly proportional to the rate of change of magnetic flux linkage.

Lenz’s Law: The electromagnetic force induced drives a current to oppose the change which produced it. This is responsible for the negative sign in Faraday’s Law.

Left hand rule and right hand rule. Much easier just to use the cross product F = q (v x B), where v is the velocity of the charges.

11.2 Power Generation and Transmission

 

11.3 Capacitance

What is a capacitor? One can think of it as a component which stores charge when a potential difference is applied across it. There are many different geometries, but the simplest is made of two parallel plates close to each other.

The most important equation for this chapter is the defining equation of capacitance

C = q / V

Unfortunately, this is not particularly informative. If we express it differently

q = CV

That is, the charge stored by the capacitor is proportional to the potential difference across it. This is analogous to a spring: the force ‘across’ a spring is proportional to its extension. The energy stored in a capacitor is

E = 1/2 C V^2

Which looks like the energy stored on a spring, 1/2 k x^2. Just as a stiffer spring requires more energy to stretch, a device with a larger capacitance requires more energy to charge.

What about capacitors in series or parallel? What is their total capacitance? Unlike resistors, we add the capacitance of capacitors in parallel, and we take the reciprocal of the sum of the reciprocals of the capacitances (1/C = 1 / C1 + 1 / C2 +…) These can be derived from Kirchoff’s Laws, basic circuit theory, and C = q / V.

Why? Capacitors in parallel act like one bigger capacitor; two pairs of parallel plates connected in parallel behave like a larger pair of parallel plates. This is like springs in parallel, the total spring constant is a sum of the individual spring constants. If you try to put a certain charge on capacitors in series, they get split among the capacitors, hence the total energy required to charge them is less. This is like extending springs in series, the extension is spread among the springs, each spring extends less, so less energy is needed in total. Consequently, the capacitance, like the spring constant, decreases when places in series. (TODO: Diagrams and example calculation)

For resistors, we have R = rho A / l. For capacitors, we have C = eps A / d. The larger the area, the more charge we can store, hence C must be directly proportional to A. The closer the plates are to each other, the more the opposite charges on either plate ‘feel’ each other, enabling more charge to be stored. Hence, C must be inversely proportional to d. The constant of proportionality turns out to be epsilon, the permittivity of the material in the gap. The permittivity measures the ability of the material to ‘concentrate’ the electric field lines. Hence, placing some material within the gap will result in a greater capacitance than just having a vacuum in the gap.

RC Circuits: Contains a resistor and a capacitor only. Use Kirchoff’s current law.

I_r = I_c

From the definition of resistance I_r = V / R

From the definition of current and capacitance I_c = dq / dt = d / dt (C/V)

Solving the differential equation, we get V = V0 exp(- t / (RC)). That is, the potential difference (across the resistor) decays exponentially. Initially, the capacitor stores a high charge, and so the pd across the resistor is higher, leading to a higher energy loss (P=V^2 / R). As the charge on the capacitor drops, the power loss decreases, hence the discharging slow down. This explains the exponential shape – the rate of decrease in voltage is proportional to the voltage. The characteristic time, the time taken for the voltage to decrease by 1/e (note that this is not the same as the half life, you can convert one to the other easily though), is thus RC. To get the equations for I and q, just use I = V / R and q = CV respectively.

When charging a capacitor, the equation is V = V0 (1 – exp(- t / (RC))). The qualitative explanation is as follows. When there’s no charge on the capacitor, charging is easy. However, if there’s already some charge there, it’s harder to put more charge on. Hence, charging slows down – the graphs flattens after a long time.

 

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