Step 1: Understanding the Question:
The image shows a series LCR circuit with potential drops across the inductor (\(V_L\)), capacitor (\(V_C\)), and resistor (\(V_R\)).
We need to find the total source voltage \(E\). Step 2: Key Formula or Approach:
In a series LCR circuit, the total voltage \(E\) is the phasor sum of the voltages across each component:
\[ E = \sqrt{V_R^2 + (V_L - V_C)^2} \text{ or } E = \sqrt{V_R^2 + (V_C - V_L)^2} \] Step 3: Detailed Explanation:
From the image:
\(V_L = 20 \text{ V}\)
\(V_C = 50 \text{ V}\)
\(V_R = 40 \text{ V}\)
Substitute these values into the formula:
\[ E = \sqrt{(40)^2 + (20 - 50)^2} \]
\[ E = \sqrt{1600 + (-30)^2} \]
\[ E = \sqrt{1600 + 900} \]
\[ E = \sqrt{2500} \]
\[ E = 50 \text{ V} \] Step 4: Final Answer:
The alternating e.m.f. is \(50 \text{ V}\).
