Question

What do you mean by root mean square of a.c. Obtain expression for ${I_{rms}}$.

Answer 1

Hint: This question requires the knowledge of the concept of alternating current and its characteristics. Alternating current is different from direct current in the way of propagation of current. It changes direction in contrast to direct current which has no direction change. Alternating current is generally used where a large quantity of current is required.

Complete step by step answer:
A current that varies its size and polarity at regular intervals is known as alternating current. It can also be defined as an electrical current that alternates or reverses its direction on a regular basis, as opposed to Direct Current (DC), which always runs in one direction.
Normally, alternating currents are accompanied by alternating voltages.

Aside from that, alternating current can readily be converted from a higher to a lower voltage level.Alternators are devices that can be used to produce or generate alternating current.Alternating current can be generated in a variety of ways, including the use of several circuits. A basic single coil AC generator, which consists of two-pole magnets and a single rectangular loop of wire, is one of the most frequent or simple means of generating AC.

The Root-Mean-Square of instantaneous current values is known as RMS. Direct current flowing through a resistance provides the RMS value of alternating current. AC has an RMS value that is higher than the average. The area covered in half-cycles can be used to calculate the RMS value of a sine current wave.
$I = {I_o}\sin wt \\
\Rightarrow dH = {I^2}Rdt \\
\Rightarrow dH = {({I_o}\sin wt)^2}Rdt \\
\Rightarrow dH = {I_o}^2R{\sin ^2}wtdt \\ $
The heat produced in time $\dfrac{T}{2}$ is given by:
$H = \int\limits_0^{\dfrac{T}{2}} {{I_o}^2R{{\sin }^2}wtdt} \\
\Rightarrow H = \dfrac{{{I_o}^2R}}{2}[\dfrac{T}{2} - 0] \\
\Rightarrow H = \dfrac{{{I_o}^2R}}{2}.\dfrac{T}{2}........(1) \\ $
We also know that,
$H = {I_{rms}}^2R.\dfrac{T}{2}.........(2)$
So, equating (1) and (2),
${I_{rms}}^2 = \dfrac{{{I_o}^2}}{2} \\
\therefore {I_{rms}} = \dfrac{{{I_o}}}{{\sqrt 2 }} = 0.707{I_o} \\ $
Hence, the rms value of ac is $0.707{I_o}$.

Note:AC is the most common type of current found in household equipment. Audio signals, radio signals, and other forms of alternating current are examples. Alternating current has a significant benefit over direct current in that it can transmit electricity across long distances with minimal energy loss.