Answer :
To find the maximum voltage [tex]\( E_m \)[/tex] across the entire RLC series circuit, we need to take into account the resistive, inductive, and capacitive components.
1. Identify Given Values:
- Capacitive reactance ([tex]\( X_C \)[/tex]): [tex]\( 8.85 \, \Omega \)[/tex]
- Inductive reactance ([tex]\( X_L \)[/tex]): [tex]\( 25.9 \, \Omega \)[/tex]
- Resistance ([tex]\( R \)[/tex]): [tex]\( 98.3 \, \Omega \)[/tex]
- Maximum voltage across the resistor ([tex]\( V_R \)[/tex]): [tex]\( 20.7 \, V \)[/tex]
2. Calculate the Net Reactance ([tex]\( X \)[/tex]):
The net reactance in the circuit can be found using the difference between the inductive and capacitive reactance:
[tex]\[
X = X_L - X_C = 25.9 \, \Omega - 8.85 \, \Omega = 17.05 \, \Omega
\][/tex]
3. Calculate the Total Impedance ([tex]\( Z \)[/tex]):
The impedance ([tex]\( Z \)[/tex]) of the circuit is determined by both the resistance and the net reactance:
[tex]\[
Z = \sqrt{R^2 + X^2} = \sqrt{98.3^2 + 17.05^2}
\][/tex]
This results in an impedance of approximately [tex]\( 99.77 \, \Omega \)[/tex].
4. Calculate the Maximum Voltage Across the Circuit ([tex]\( E_m \)[/tex]):
Ohm’s Law tells us that the voltage across a component is equal to the current through it times its impedance. Since the circuit is series, the current is the same through all components. Therefore, the ratio of maximum voltage across any component to its impedance is the same across the entire circuit. So:
[tex]\[
E_m = V_R \times \left( \frac{Z}{R} \right) = 20.7 \, V \times \left( \frac{99.77 \, \Omega}{98.3 \, \Omega} \right)
\][/tex]
This results in a maximum voltage across the circuit of approximately [tex]\( 21.01 \, V \)[/tex].
So, the maximum voltage [tex]\( E_m \)[/tex] across the entire RLC circuit is approximately 21.01 V.
1. Identify Given Values:
- Capacitive reactance ([tex]\( X_C \)[/tex]): [tex]\( 8.85 \, \Omega \)[/tex]
- Inductive reactance ([tex]\( X_L \)[/tex]): [tex]\( 25.9 \, \Omega \)[/tex]
- Resistance ([tex]\( R \)[/tex]): [tex]\( 98.3 \, \Omega \)[/tex]
- Maximum voltage across the resistor ([tex]\( V_R \)[/tex]): [tex]\( 20.7 \, V \)[/tex]
2. Calculate the Net Reactance ([tex]\( X \)[/tex]):
The net reactance in the circuit can be found using the difference between the inductive and capacitive reactance:
[tex]\[
X = X_L - X_C = 25.9 \, \Omega - 8.85 \, \Omega = 17.05 \, \Omega
\][/tex]
3. Calculate the Total Impedance ([tex]\( Z \)[/tex]):
The impedance ([tex]\( Z \)[/tex]) of the circuit is determined by both the resistance and the net reactance:
[tex]\[
Z = \sqrt{R^2 + X^2} = \sqrt{98.3^2 + 17.05^2}
\][/tex]
This results in an impedance of approximately [tex]\( 99.77 \, \Omega \)[/tex].
4. Calculate the Maximum Voltage Across the Circuit ([tex]\( E_m \)[/tex]):
Ohm’s Law tells us that the voltage across a component is equal to the current through it times its impedance. Since the circuit is series, the current is the same through all components. Therefore, the ratio of maximum voltage across any component to its impedance is the same across the entire circuit. So:
[tex]\[
E_m = V_R \times \left( \frac{Z}{R} \right) = 20.7 \, V \times \left( \frac{99.77 \, \Omega}{98.3 \, \Omega} \right)
\][/tex]
This results in a maximum voltage across the circuit of approximately [tex]\( 21.01 \, V \)[/tex].
So, the maximum voltage [tex]\( E_m \)[/tex] across the entire RLC circuit is approximately 21.01 V.
