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General Physics 2
Example Sheet 2
1. *An electron moves in a circular path with radius 𝑟 = 4.00 cm in the space between two
concentric cylinders. The inner cylinder is positively charged with radius 𝑎 = 1.00 mm
and the outer cylinder is negatively charged with radius 𝑏 = 5.00 cm. The potential
difference between the two cylinders is 𝑉 = 120 V. There is a uniform magnetic field
𝐵 = 1.3 × 10−4 T into the page.
𝑣
𝑏
𝑟
𝑎
The electric field in the space between the cylinders is give by 𝐸 𝑟 = 𝑉 𝑟 ln 𝑏 𝑎 .
Calculate the speed 𝑣 of the electron.
2. *In the electron gun of a TV picture tube, the electrons (charge 𝑒, mass 𝑚) are accelerated
by a voltage 𝑉. After leaving the electron gun, the electron beam travels a distance D to the
screen; in this region there is a transverse magnetic field of magnitude 𝐵.
a) Show that the deflection of the beam due to the magnetic field is
𝐵𝐷2
𝑒
𝑑 =
.
2
2𝑚𝑉
b) For 𝑉 = 750 V, 𝐷 = 50 cm and 𝐵 = 5.0 × 10−5 T, evaluate 𝑑. Is this deflection
significant?
3. *A semicircle-shaped conducting wire of radius 𝑅 is placed in a uniform magnetic field 𝐵
perpendicular to the plane of the wire. The current 𝐼 is passed through the wire. Find the
resultant magnetic force on the wire.
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4. * In Figure 1, a circular loop of radius 𝑅 carries current 𝐼.
Figure 1
Figure 2
a) By using Biot-Savart law, show that the magnetic field at distance 𝑥 from the center
along the symmetry axis in Fig. 1 is given by
𝜇0 𝐼𝑅 2
𝐵 =
.
2 𝑥 2 + 𝑅 2 3/2
b) Figure 2 shows a Helmhotz coil consisting of two circular coils, each having 𝑁 turns,
on the same axis. The separation between the coils is equal to the radius 𝑅 of each
coil. Show that the magnetic field at the center is given by
8 𝜇0 𝑁𝐼
𝐵 =
.
5 5 𝑅
5.
*The current 𝐼 in a long straight wire is upward. There is a
rectangular loop with 𝑎 = 12 cm and 𝑏 = 24 cm at distance
𝑑 = 12 cm from the wire. If the current in the wire increases
at rate 𝑑𝐼 𝑑𝑡 = 9.6 As-1, calculate the induced emf in the
loop.
6. *A conducting rod with mass 𝑚 and length 𝐿 moves on two parallel friectionless rails in a
uniform magnetic field 𝐵 into the page. The two rails are connected via a resistor whose
resistance is 𝑅. The bar is moving to the right.
a) When the bar is moving at speed 𝑣 to the right, show
that the equation of motion of the bar is
𝑑𝑣
𝐵 2 𝐿2 𝑣
𝑚
= −
.
𝑑𝑡
𝑅
b) If the initial speed of the bar is 𝑣𝑖 , find the speed 𝑣(𝑡)
as a function of time 𝑡.
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7. *Consider a toroid with rectangular cross-section, closely wound with 𝑁 turns. The inner
and the outer radii are 𝑎 and 𝑏 respectively. The height of the coil is 𝐻.
Show that the inductance of this toroid is given by
𝜇0 𝑁 2 𝐻
𝑏
𝐿 =
ln
.
2𝜋
𝑎
8. An LR circuit shown in the figure below contains a resistor 𝑅1 and an inductance 𝐿 in series with a
battery of emf ℰ . The switch 𝑆 is initially closed for a long time before 𝑡 = 0. At 𝑡 = 0, the switch 𝑆
is opened, so that an additional resistance 𝑅2 is now in series with the other elements.
a) Write down the initial current 𝐼0 in the circuit at 𝑡 = 0.
Assume that battery emf ℰ is negligible compared with the total emf around the circuit
just after the switch is open.
b) By integration, find 𝐼 𝑡 in terms of 𝐼0 , 𝑅1 , 𝑅2 and 𝐿.
c) Show that 𝑅2 ≫ 𝑅1 .
9. The Bridge circuit below is balanced when the variable resistor and capacitor are adjusted
to 9 kΩ and 1 𝜇F respectively.
Find the value of 𝑅 and 𝐿 in the circuit.
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10. *Consider a circuit below. A voltage source 𝑉in = 𝑉0 𝑒 𝑖𝜔𝑡 is applied to the input.
a) Find the expression for 𝑉out 𝑉in .
b) The input angular frequency 𝜔 is varied. Sketch the amplitude and the phase of 𝑉out
as the function of 𝜔
c) For 𝑉0 = 10 V, 𝜔 = 300 rad s-1, 𝑅 = 10 Ω, 𝐿 = 10 mH and 𝐶 = 100 𝜇F. Calculate the
average power (𝑃rms ) across the output terminals.
11. *Consider the circuit shown below.
a) Find the total complex impedance.
b) If 𝐿 = 𝐶𝑅 2 and the voltage source has angular frequency 𝜔 = 1 𝐿𝐶 . Show that the
current flowing through the circuit is 𝜋 4 out of phase with the applied voltage.
Which leads?