The reactance, X, of a capacitor is given by the relationship: [latex] X=\frac{1}{2 \pi fC} [/latex] Find the value of capacitance that will exhibit a reactance of 10 kΩ at a frequency of 400 Hz.

First, let’s summarize what we know: X = 10 kΩ f = 400 Hz π = 3.142 (or use the ´π´ button on your calculator) C = ? We need to re-arrange the formula to make C the subject. This is done as follows: [latex] X=\frac{1}{2 \pi fC} [/latex] Cross multiplying gives: [latex] C=\frac{1}{2 \pi f \times X} [/latex] (note that we have just ‘swapped’ the C and the X over). Next, replacing π, C and f by the values that we know gives: [latex] C=\frac{1}{2 \times 3.142 \times400 \times 10\times10^3} =\frac{1}{25,136 \times10^3} =\frac{1}{2.5136 \times10^7}=\frac{1}{2.5136} \times10^{-7}=0.398 \times10^{-7} \ F [/latex] Finally, it would be sensible to express the answer in nF (rather than F). To do this, we simply need to multiply the result by [latex] 10^9 [/latex] , as follows: [latex] C = 0.398 × 10^{−7}× 10^9 = 0.398 × 109^{9−7} = 0.398 × 10^2 = [/latex] 39.8 nF

Question A8.9: The reactance, X, of a capacitor is given by the relationshi...

Electronic Circuits Fundamentals and Applications

The reactance, X, of a capacitor is given by the relationship:

X=\frac{1}{2 \pi fC}

Find the value of capacitance that will exhibit a reactance of 10 kΩ at a frequency of 400 Hz.

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First, let’s summarize what we know:

X = 10 kΩ
f = 400 Hz
π = 3.142 (or use the ´π´ button on your calculator)
C = ?

We need to re-arrange the formula to make C the subject. This is done as follows:

X=\frac{1}{2 \pi fC}

Cross multiplying gives:

C=\frac{1}{2 \pi f \times X}

(note that we have just ‘swapped’ the C and the X over).
Next, replacing π, C and f by the values that we know gives:

C=\frac{1}{2 \times 3.142 \times400 \times 10\times10^3} =\frac{1}{25,136 \times10^3} =\frac{1}{2.5136 \times10^7}=\frac{1}{2.5136} \times10^{-7}=0.398 \times10^{-7} \ F

Finally, it would be sensible to express the answer in nF (rather than F). To do this, we simply need to multiply the result by $10^9$ , as follows:
$C = 0.398 × 10^{−7}× 10^9 = 0.398 × 109^{9−7} = 0.398 × 10^2 =$ 39.8 nF