RMS Current Formula Shortcut

 

❓ Question

The current in a circuit varies as

$$i=i_0\left(\frac{t}{T}\right)$$

for $0 \le t \le T$.

Find the RMS value of the current over the interval $0$ to $T$.

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✍️ Formula for RMS Current

$$i_{\text{rms}} = \sqrt{\frac{1}{T}\int_0^T i^2\,dt}$$

Substitute

$$i=i_0\frac{t}{T}$$
$$i_{\text{rms}} = \sqrt{\frac{1}{T}\int_0^T \left(i_0\frac{t}{T}\right)^2 dt}$$
$$= \sqrt{\frac{i_0^2}{T^3}\int_0^T t^2\,dt}$$

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🔷 Evaluate the Integral

$$\int_0^T t^2\,dt = \left[\frac{t^3}{3}\right]_0^T = \frac{T^3}{3}$$

Therefore,

$$i_{\text{rms}} = \sqrt{\frac{i_0^2}{T^3}\cdot\frac{T^3}{3}}$$
$$= \sqrt{\frac{i_0^2}{3}}$$
$$= \frac{i_0}{\sqrt3}$$

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✅ Final Answer

$$\boxed{i_{\text{rms}}=\frac{i_0}{\sqrt3}}$$

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🔷 JEE Shortcut

For a linearly increasing current

$$i=I_{\max}\frac{t}{T}$$

over one interval $0\to \mathrm{T,}$

$$\boxed{i_{\text{rms}}=\frac{I_{\max}}{\sqrt3}}$$

Directly use:

$$\boxed{\frac{i_0}{\sqrt3}}$$