Tutorial 4B - Reactance of a Capacitor

Learning Objectives

To understand the concept of reactance in a capacitor.

To explain how the reactance decreases with increasing frequency.

To recognise and use the equation:

Key Questions

How do we explain how a capacitor allows AC to flow?

What is reactance?

How do we measure reactance?

Reactance of a Capacitor

If we connect a capacitor in series with a bulb and change the frequency, we notice that:

1.      At low frequencies, the bulb in series with the capacitor glows;

2.      As we increased the frequency, the bulb became brighter.

3.      It allowed AC to pass.

 

It was as if the capacitor had a kind of “resistance”.

 

Now we know that anything with a resistance allows current to flow, but a capacitor cannot allow current to flow because it has a layer of insulating material between the plates.  Without that insulation, the capacitor would be useless.  The resistance of all capacitors can be considered to be infinite.

 

So how did the bulb light up?  Consider a capacitor that has a capacitance C, charged up to a voltage V, and has a charge Q on it.  We know that current is given by the equation:

The triangular symbol D is “Delta”, a Greek upper-case letter ‘D’.  It is the physics code for “change in”.

 

For an instantaneous value of current, we can say that:

Since Q = CV, we can also write:

The point of this is that if the frequency is low, the instantaneous current is low.  If the frequency is increased, the time interval, dt, gets smaller.  As it’s the denominator (i.e. downstairs in the fraction), the instantaneous current gets bigger.  So the bulb lights up as the frequency increases.

 

Reactance

This brings us on to the reactance of a capacitor.  Reactance in the simplest terms is the equivalent resistance that results from a given frequency.  Reactance is more formally defined as:

 

the ratio of the potential difference to the current in a capacitor circuit.

 

Note that the phrase “flowing through” is not used.

 

Reactance has the physics code XC and has the units Ohms (W).  So we can write an equation:

 

 

However we haven’t mentioned the frequency, f.  We can also refer to the angular velocity, w, which is directly proportional to the frequency.  We know that the reactance is frequency dependent.

 

 

 

  Remember that current does not flow through a capacitor.  There is an insulating layer.

Question 6

   Discuss whether a capacitor would allow a constantly changing unidirectional current to pass.

Answer

Question 7

When we talk about the definition of reactance, we do not use the idea of a current flowing through the capacitor.  Why not?

Answer

Question 8

How do we know that reactance is frequency dependent?

Answer

 

Equation for the reactance of a capacitor

Look at this circuit:

For a capacitor of capacitance C connected to a voltage source with a frequency f, the reactance, XC is given by:

 

 

 

If you want to see the derivation of this equation, please click HERE to go to the extension.  Or you could press the Extension button on the bottom of this page.

 

Question 9

       A 10 mF capacitor has a reactance of 320 W

       (a) Show that the frequency of the AC supply is about 50 Hz;

  (b) Calculate the reactance at 1000 Hz.

Answer

 

If we plot reactance against frequency, the graph is curved.  It is a hyperbola, showing a y = 1/x relationship:

 

 

If we plot reactance against 1/frequency, we should get a straight line, going through the origin:

 

 

This graph was plotted using data from a real experiment.  The dotted line shows an idealised straight line, but the data are not that far off the ideal.

 

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