Question 1a

Marks: 4

A thundercloud is 2.20 km above the surface of Earth. The charge on the base of the cloud is −30.0 C. The air between the cloud and the Earth is humid and rainy, making it 4.35 % water by mass.

thundercloud

The relative permittivity for a homogenous mixed medium, ε_r(m), is given by:

$ε_{r (m)} = ϕ_{1} ε_{r (1)} + ϕ_{2} ε_{r (2)}$

Where ε_r(1)& ε_r(2) represent the relative permittivities of each material in the mixture and Φ₁ & Φ₂ represent the fraction by mass of each material.

The permittivity of water is 7.08 × 10⁻¹⁰ F m⁻¹and the permittivity of air is 8.85 × 10⁻¹² F m⁻¹.

(a)

Show that the dielectric constant of the rainy air is about 4.

[4]

Question 1b

Marks: 3

The cloud has a roughly rectangular base of length 5.00 km and the potential difference from the base of the cloud to Earth is −8.00 × 10⁸ V.

(b)

Calculate the width of the cloud.

[3]

Question 1c

Marks: 4

Lightning strikes a tree after strong winds increase the potential difference of the system to −9.0 × 10⁹ V. Air conducts electricity once there is a potential difference of 3.00 × 10⁶ V per metre.

(c)

Given that, in a storm, the rainy air has a resistance of 136 Ω m⁻¹, determine the time period of the lightning strike.

Assume the cloud's area and distance from the ground are unchanged by the wind.

[4]

Question 2a

Marks: 3

A tattoo removal company uses a pulsed Nd:YAG laser used a capacitor to store energy and produce a short laser pulse. When pulses in the nanosecond range are discharged, tattoo pigments are removed without damaging skin cells.

The research and design team of a company that produces these lasers are experimenting with different dielectrics in the capacitor. One suggestion is to use a cubic dielectric material that is half glass, with permittivity ε_G, and half silicon, with permittivity ε_Si, split down the middle.

This mixed medium is used in two orientations during tests.

capacitor-comparison

The capacitor with the dielectric split vertically has a capacitance of C₁ and the capacitor with the dielectric split down the middle has a capacitance of C₂.

(a)

Write an expression for capacitance, C₁, in terms of L, ε_G and ε_Si.

[3]

Question 2b

Marks: 4

(b)

Write an expression for capacitance, C₂, in terms of L, ε_G and ε_Si.

[4]

Question 2c

Marks: 6

Capacitor 1 and capacitor two are connected in two different circuits. The charge on capacitor 1 is twice that of capacitor 2.

The dielectric constant of the silicon is 4.3 and the relative permittivity of the glass is 6.5

(c)

Show that the energy stored in capacitor 2 is 26 % of the energy stored in capacitor 1.

(i)

Show that

\frac{E_{2}}{E_{1}} = \frac{{(ε_{S i} + ε_{G})}^{2}}{16 ε_{G} ε_{S i}}

, where E_n is the energy stored in capacitor n and ε_z is the permittivity of material z.

[3]

(ii)

Calculate E₂ as a percentage of E₁.

[2]

Question 3a

Marks: 4

A capacitor with capacitance 220 μF is attached to an ac voltage source which provides a voltage which varies according to the following equation:

$V = V_{0} \sin (ω t)$

The rate of change of the voltage is given by:

$\frac{∆ V}{∆ t} = ω V_{0} \cos (ω t)$

This circuit is called ‘purely capacitive’, meaning that resistance can be assumed to be zero.

ib-hl-hard-sq-3a-question

The power supply is adjusted such that the initial voltage is 6.0 V and the alternating voltage frequency is 4000 rad s^–1.

(a)

Calculate the magnitude of the current flowing in the circuit at time t = 3.14 s.

[4]

Question 3b

Marks: 3

The variation of voltage V and current I in the circuit is shown.

q5b-fig-2

(b)

Discuss the phase difference between the variation of V and I in the circuit.

[3]

Question 3c

Marks: 4

'Capacitive reactance' X is the opposition to current flow in a purely capacitive circuit as described in part (a). Capacitive reactance is comparative to resistance, in that it is measured by the same units Ω.

(c)

By considering the ratio of the maximum voltage and current in the circuit, show that the capacitive reactance X is given by

X = \frac{1}{ω C}

and verify that X has the same units as resistance.

[4]

Question 4a

Marks: 2

A vacuum capacitor is connected, along with a resistor, to a cell with an emf of 12 V in the configuration below. In a vacuum, the capacitor has a capacitance of 4.5 μF.

capacitor

Initially, the vacuum capacitor is uncharged. At a time of t = 0 s, the switch is placed at position A. The voltage across the capacitor is recorded over time and plotted in the graph below.

11-3-4a_2

(a)

Sketch a second line on the axes, showing the variation in the voltage across the resistor over time.

[2]

Question 4b

Marks: 3

(b)

Using the graph, calculate the resistance, R, of the resistor.

[3]

Question 4c

Marks: 3

The vacuum chamber is now filled with acetone, which has a dielectric constant of 19.5.

(c)

Calculate the new charge stored in the capacitor when the voltage across the capacitor is half its maximum value.

[3]

Question 4d

Marks: 3

(d)

Once the capacitor is fully charged, the switch in the diagram in part (a) changes to position B.

(i)

Describe the energy changes in the capacitor.

[1]

(ii)

Explain why these energy changes occur.

[2]

DP IB Physics: HL

Topic Questions

11.3 Capacitance

Question 1a

Question 1b

Question 1c

Question 2a

Question 2b

Question 2c

Question 3a

Question 3b

Question 3c

Question 4a

Question 4b

Question 4c

Question 4d

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