Energy stored by a capacitor

The circuit shows a 1 mF capacitor connected to an a.c. supply. The graph shows how the potential difference V varies with time t.

Calculate the root‑mean‑square potential difference.

The formula used to generate this graph is V = 5 sin (100 πt)

Explain why this formula leads to the graph above.

A spreadsheet is used to model how the current I in the 1 mF capacitor varies with t. Six rows of the spreadsheet are shown below.

Explain how cell E10 has been calculated.

State the formula used to calculate cell G11.

Calculate the maximum energy stored on the capacitor.

The spreadsheet data are used to plot a graph to show how I varies with t. This is shown as a dashed line below.

The corresponding graph of V against t is also shown as a continuous line.

Deduce whether the capacitor dissipates power over one cycle of the a.c. supply.

*5 The diagram shows a ‘shaker torch’. When the torch is shaken, a strong magnet moves forwards and backwards through a copper coil, powering a light-emitting diode (LED).

Each time the magnet moves through the coil a current pulse is generated. The coil is connected to a capacitor via a diode, as shown.

Once the torch has been shaken for a few minutes the switch is closed and the LED lights for a short while.

Explain how the shaker torch is able to light the LED.

A capacitor is connected to a power supply and charged to a potential difference V₀.

The graph shows how the potential difference V across the capacitor varies with the charge Q on the capacitor.

At a potential difference V₀ a small charge ∆Q is added to the capacitor. This results in a small increase in potential difference ∆ V across the capacitor.

Which of the following gives the approximate increase in energy stored on the capacitor due to this extra charge?

The properties of capacitors make them useful in timing circuits.

The following circuit is used to provide an input Y to an integrated circuit.

Initially the capacitor is uncharged. The switch S₁ is closed. Sketch a graph to show how the potential at point Y varies with time.

When the potential at Y is 8.0 V, the switch S₂ is closed.

Calculate the time taken for the potential at Y to decrease to 2.0 V.

Calculate the energy stored on the capacitor when the potential at Y is 2.0 V.

When the potential at Y is 2.0 V, the switch S₂ is opened. Calculate the power dissipated by the resistance R₁ when the potential at Y is 2.0 V.

The properties of capacitors make them useful in timing circuits.

The following circuit is used to provide an input Y to an integrated circuit.

Initially the capacitor is uncharged. The switch S₁ is closed. Sketch a graph to show how the potential at point Y varies with time.

When the potential at Y is 8.0 V, the switch S₂ is closed.

Calculate the time taken for the potential at Y to decrease to 2.0 V.

Calculate the energy stored on the capacitor when the potential at Y is 2.0 V.

When the potential at Y is 2.0 V, the switch S₂ is opened. Calculate the power dissipated by the resistance R₁ when the potential at Y is 2.0 V.

A ‘Gauss gun’ can be made from five ball bearings of equal mass and two magnets, as shown.

Pairs of ball bearings are placed to the right of two strong magnets. A single ball bearing is released from the left, as shown. The ball bearing is attracted to, and collides with, the first magnet. This and all subsequent collisions can be assumed to be elastic.

Explain what happens to make the last ball bearing on the right subsequently move off with a large velocity.

A student set up the apparatus shown to measure the speed of the last ball bearing. The ‘Gauss gun’ was placed at the end of a bench, so that the ball bearing left the gun and broke two strips of metal foil which formed part of an electric circuit.

As the ball bearing left the gun, it broke the first foil strip at its centre so that the capacitor started to discharge. When the ball bearing broke the second foil strip the capacitor discharge stopped.

Calculate the energy stored in the capacitor when it was fully charged.

The voltmeter reading halved in the time taken for the ball bearing to travel between the two foil strips.

Show that the time taken for the ball bearing to travel between the two foil strips was about 0.1 s.

The two foil strips were 0.50 m apart.

Calculate the horizontal velocity of the ball bearing.

The student positioned the second foil strip with its centre 8.0 cm lower than the centre of the first foil strip.

Deduce whether the ball bearing broke the second foil strip at its centre.

Assume the ball bearing was travelling horizontally as it broke the first foil strip.

A capacitor of capacitance C is charged to a potential difference V by a power supply. The energy stored on the charged capacitor is W. What would be the energy stored if the potential difference were 2V?