Capacitor reactance:
formula and worked example

What capacitive reactance describes

A resistor opposes current the same way regardless of frequency. A capacitor does not. It stores charge, and how much alternating current flows depends on how quickly the voltage is switching. At DC the capacitor stops the current once it charges; at high frequency, the voltage reverses so fast that current flows almost freely.

Capacitive reactance, written Xc, expresses this as an opposition in ohms. It behaves like resistance for working out current magnitude, but with a key difference: it stores and returns energy instead of turning it into heat, and the current leads the voltage by a quarter of a cycle.

• Frequency: the AC frequency in hertz (50 Hz is common for mains in India and many countries).
• Capacitance f: the capacitance in farads. A 10 µF capacitor is 0.00001 F.
• Result: reactance in ohms.

The formula and its two levers

Both frequency and capacitance sit in the denominator, so Xc is inversely proportional to each. Double either one and the reactance halves. Because they appear as a product, you can trade one for the other: a 1 µF capacitor at 500 Hz offers the same opposition as a 10 µF at 50 Hz.

At DC the frequency is zero and the denominator vanishes, which is the mathematical way of saying the opposition is infinite: an ideal capacitor blocks steady current entirely.

Worked example: 10 µF at 50 Hz

A 10 µF capacitor sits on a 50 Hz supply, such as a motor run capacitor or a mains filter element. What opposition does it present?

1. 1Convert the capacitance: 10 µF = 10 × 10⁻⁶ = 0.00001 F.
2. 2Compute 2 × π × 50 × 0.00001 = 0.0031416.
3. 3Take the reciprocal: 1 ÷ 0.0031416 = 318.31 Ω.
4. 4Find the current at 230 V: 230 ÷ 318.31 ≈ 0.72 A, ignoring any series resistance.

The 0.72 A is reactive current. It flows back and forth with the supply and dissipates almost no real power, which is why capacitors can pass amperes without getting hot.

Reading the number against the frequency

The same capacitor looks very different across frequencies, which is what makes capacitors useful for separating signals. At 1 kHz, this 10 µF part has a reactance of only about 15.9 Ω; at 100 Hz it is about 159 Ω; at 50 Hz it is 318 Ω.

• Coupling: choose a capacitor whose reactance at the lowest signal frequency is small compared with the load, so the signal passes unharmed.
• Filtering and decoupling: choose a value with low reactance at the noise frequency, so unwanted ripple is shunted to ground.
• Power-factor and run capacitors: the capacitor's reactance sets the leading current that offsets lagging motor current.

Real capacitors also have series resistance and inductance, so beyond some frequency the part stops behaving like the ideal formula, and a small ceramic capacitor can work better than a large electrolytic for high-frequency noise.

How frequency and capacitance shift the answer

Raising the frequency by 10%, from 50 to 55 Hz, lowers the reactance from 318.31 Ω to 289.37 Ω, a fall of about 9.1%, not 10%. The reciprocal relationship means that an increase of one tenth reduces the result by a ninth of the original, which is a smaller percentage than the input rise.

For capacitance the same logic holds: a 10% larger capacitor gives about 9.1% less reactance. This is also why a capacitor tolerance of ±20%, common for electrolytics, turns into a spread of reactance of roughly -17% to +25% around the nominal.

Mistakes with capacitor reactance

• Entering microfarads as farads, which makes the answer a million times too small and is the most common slip here.
• Using the formula at DC, where the answer is not defined and the real behaviour depends on leakage.
• Adding reactance to resistance directly; in series, the magnitudes combine as a square root of the sum of squares, not as a plain sum.
• Assuming the capacitor's labelled value holds under all conditions, when electrolytics can lose a large share of their capacitance with age and heat.

A scenario: choosing a coupling capacitor

A stage with an input resistance of 10 kΩ must pass signals down to about 20 Hz. The usual guide is to pick a capacitor whose reactance at that frequency equals the resistance, which puts the corner there. C = 1 ÷ (2π × 20 × 10,000) = 0.8 µF, so a 1 µF capacitor is the nearest common choice and the corner sits near 16 Hz.

At that corner the signal is down by about 3 dB, so many designers go one value higher to keep the low end clean. Tolerance and the capacitor's voltage coefficient make a safe margin worthwhile.