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# Time Constant and Energy Stored in Capacitors

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## Time Constant and Energy Stored in Capacitors

**Capacitors discharge exponentially.** That means that their charge falls away in a similar way to radioactive material decay. In radioactivity you have a half-life, in capacitance you have a **'time constant'.**

**The rate of removal of charge is proportional to the amount of charge remaining.**

As time steps forward in equal intervals, T (called the **time constant**), **the charge drops by the same proportion each time**. It turns out that each for interval, T, the charge or current drops to about 0.37 (37%) of its initial value. (** Note:** For the mathematicians amongst you, this number can be calculated using 1/e, where e is the exponential constant with a value of 2.718.)

**We can calculate the time constant, T using the equation:**

**T = RC**

*Where:*

**T **= time constant

**R **= resistance in the circuit (Ω)

**C **= capacitance of the circuit (F)

So the factor that governs how quickly the charge drops is a combination of the capacitance of the capacitor and the resistance it is discharging through.

In practice it takes **0.69 x RC** (ln2 x RC) for the charge to be half its original value. In this time the discharge current also drops to half its original value too.

**To calculate the charge left, Q, on a capacitor after time, t, you need to use the equation:**

*Where:*

**Q _{0}** = initial charge on the capacitor

**Q** = charge on the capacitor at any time

**t** = time

**RC** = time constant

**Likewise the current or voltage at any time can be found using:**

As all of these relationships are exponential, natural log graphs can be drawn to obtain values for the time constant. **For instance:**

(Remeber for y = mx + c

m gives the gradient of the graph

c is the intercept on the y axis when x = 0)

The potential difference across the plates of a capacitor is directly proportional to the charge stored on the plates. This gives a straight line through the origin on a voltage-charge graph. **The area under this graph gives the energy stored in a capacitor.**

As the area under the graph is a triangle,

area = ½ base x height.

** Note:** the energy used by the cell to charge the capacitor, W = QV, but the energy stored on the capacitor = 1/2 QV. So half the energy is lost in the circuit as heat energy as the capacitor is changed.

As capacitors are able to store energy, they can be used in back-up systems in electrical devices, such as computers.