The voltage across the ends of a resistor is wtVv sin0= . The r.m.s.value of this voltage is :

To find the root mean square (r.m.s) value of the voltage expressed as $V_v(t) = V_m \sin(\omega t)$, where $V_m$ is the maximum voltage (amplitude), we will follow the steps below.

1. ### Break Down the Problem

We need to derive the r.m.s. value of a sinusoidal voltage function:

• Identify the maximum voltage $V_m$.
• Apply the formula for r.m.s. value for sinusoidal waves.

2. ### Relevant Concepts

The r.m.s value of a sinusoidal current or voltage is given by the formula:

$V_{r.m.s} = \frac{V_m}{\sqrt{2}}$

3. ### Analysis and Detail

1. Given:
   • The sinusoidal voltage is $V_v(t) = V_m \sin(\omega t)$.
2. The r.m.s value is calculated as follows:
   • Substitute $V_m$ into the formula: $V_{r.m.s} = \frac{V_m}{\sqrt{2}}$

4. ### Verify and Summarize

The formula for r.m.s voltage derived from the definition is consistent and widely used for sinusoidal functions. Thus the calculation is verified as correct.

Final Answer

The r.m.s. value of the voltage across the resistor is:

$V_{r.m.s} = \frac{V_m}{\sqrt{2}}$