Q. The electric current in the circuit is given as

i = i₀ ( t / T )

The r.m.s current for the period t = 0 to t = T is ________.

Correct Answer: i₀ / √3

Solution

The r.m.s value of a time varying current over a time period T is defined as:

i_rms = √ [ (1 / T) ∫₀^T i² dt ]

Given current is:

i = i₀ ( t / T )

Squaring the current:

i² = i₀² ( t² / T² )

Substituting in the r.m.s formula:

i_rms = √ [ (1 / T) ∫₀^T i₀² ( t² / T² ) dt ]

Taking constants outside the integral:

i_rms = i₀ √ [ (1 / T³) ∫₀^T t² dt ]

Integral of t² from 0 to T is:

∫₀^T t² dt = T³ / 3

Substituting back:

i_rms = i₀ √ ( 1 / 3 )

i_rms = i₀ / √3

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