Assignment 3
1. A parallel RLC circuit driven by an AC power supply with amplitude Es at driving frequency of
𝜔 is shown in figure 1.
a. What is the potential drop across resistor, inductor and capacitor the moment the
capacitor is fully charged?
b. If the current flowing through the inductor is smaller than that into the capacitor,
draw the representative phasor diagram and indicates IR, IL IC, and IS.
c. What is the current amplitude and impedance of the circuit?
Figure 1
2. A transformer with input AC voltage of amplitude V1 and deliver and output voltage of
amplitude V2.
a. Ideally, the magnetic flux passes through all turns of primary and of the secondary.
The magnetic flux due to primary coil is given by Φ1 = I1 L1 + 𝑀I2 = 𝑁1 Φ. The
magnetic flux due to coil secondary coil is Φ2 = I2 L2 + 𝑀I1 = 𝑁2 Φ. In this case
show that M2 = L1L2, where M is the mutual inductance of the coils and L1, L2 are
their individual self-inductances.
Suppose the secondary coil is connected to a resistor R
b. If the primary coil is driven by AC voltage V𝑖𝑛 = V1 cos⁡(𝜔𝑡), show that the two
currents satisfy the relations: L1
𝑑I1
𝑑𝑡
+𝑀
𝑑I2
𝑑𝑡
= V1 cos⁡(𝜔𝑡); L2
𝑑I2
𝑑𝑡
+𝑀
𝑑I1
𝑑𝑡
= −I2 𝑅
c. Using the result in (a), solve these equations for I1(t) and I2(t). (assuming I1 has no DC
component).
d. Show that the output voltage (Vout = I2R) divided by the input voltage (Vin) is equal to
the L2 divided by the M2.
e. Calculate the input power (Pin = VinI1) and the output power (Pout = VoutI2), and show
that their averages over a full cycle are equal: ⟨𝑃𝑖𝑛 ⟩ = ⟨𝑃𝑜𝑢𝑡 ⟩ =
Figure 2
𝑉12 𝐿2
.
2𝐿1 𝑅
3. A circuit shown in Figure 3 has long been connected to a battery. At t = 0, switch S is thrown,
bypassing the battery
a. What is the current at any subsequent time t?
b. What is the total energy delivered to the resistor?
c. Show that the energy delivered to the resistor is equal to the energy originally
stored in the inductor.
Figure 3
4. An alternating current 𝐼 = 𝐼𝑜 cos⁡(𝜔𝑡) flows through a long straight conducting wire and
returns along a coaxial conducting tube of a radius b.
a. Assuming the field goes to zero at infinity distance. Find the electric field E (r,t).
b. What is the total displacement current 𝐼𝑑 ?
c. What is the ratio of 𝐼𝑑 over I (𝐼𝑑 /𝐼)?
5. A capacitor is connected to a thin wire and charged with a constant current I, as shown in
figure 4. The radius of the capacitor is a and the separation of the plates is w << a. Assuming
the current flows out over the plates in such a way that the surface charge is uniform at any
given time and is zero at t = 0.
a. Find the electric field between the plates as a function of time t.
b. Find the displacement current through a circle of radius s in the plane midway
between the plates.
c. Find the magnetic field at a distance s from the axis.
Figure 4