A direct current of $5\, A$ is superposed on an alternating current $I = 10 \,\sin \omega t$ flowing through the wire. The effective value of the resulting current will be

Question

In an a.c. circuit, voltage and current are given by: \[ V = 100 \sin(100t) \, \text{V} \quad \text{and} \quad I = 100 \sin\left(100t + \frac{\pi}{3}\right) \, \text{mA}. \] The average power dissipated in one cycle is:

Answer 1

To determine the effective value of the given current, we need to calculate the root mean square (RMS) value of the total current. The total current includes both a direct current (DC) component and an alternating current (AC) component.

The given components are:

Direct current (DC), I_{\text{DC}} = 5 \, A
Alternating current (AC), I_{\text{AC}} = 10 \sin \omega t

The RMS value of the total current can be calculated using the formula:

I_{\text{rms}} = \sqrt{I_{\text{DC}}^2 + I_{\text{AC(rms)}}^2}

First, calculate the RMS value of the AC component:

The RMS value of an AC current I = I_0 \sin \omega t is given by:

I_{\text{AC(rms)}} = \frac{I_0}{\sqrt{2}}

where I_0 is the peak value of the AC current.

Here, I_0 = 10 \, A, so:

I_{\text{AC(rms)}} = \frac{10}{\sqrt{2}} = \frac{10 \times \sqrt{2}}{2} = 5 \sqrt{2} \, A

Substituting these values back into the formula for I_{\text{rms}}:

I_{\text{rms}} = \sqrt{(5)^2 + (5\sqrt{2})^2}

= \sqrt{25 + 50}

= \sqrt{75}

= 5\sqrt{3} \, A

Therefore, the effective value of the resulting current is 5\sqrt{3} \, A, which matches the given correct answer.

Answer 2