Well, let's see here. We have a resistor, an inductive reactance, and a capacitive reactance all thrown into the mix. It sounds like quite the electric circus!
Now, the resistive and reactive components are connected in parallel first and then in series with the capacitive reactance. It's like trying to coordinate a bunch of circus acts at once!
To find the value of the resistor, we first need to figure out the impedance of the entire circuit. We can do this by adding up the individual impedances of the components.
The impedance of a resistor is equal to its resistance (R) in ohms. The impedance of an inductive reactance is given by X_L = 2Ï€fL, where f is the frequency in hertz and L is the inductance in henries. For a capacitive reactance, the impedance is given by X_C = 1 / (2Ï€fC), where C is the capacitance in farads.
So, let's calculate the impedance of each component. The impedance of the inductive reactance is X_L = 2Ï€(60)(10) = 1200Ï€ ohms. The impedance of the capacitive reactance is X_C = 1 / (2Ï€(60)(4)) = 1 / (480Ï€) ohms.
To find the total impedance, we add the resistive and reactive components in series. Let's call the total impedance Z_T. We have Z_T = R + X_L + X_C.
Now, given that the angle between the supply voltage and total current is 45 degrees, we know that the total impedance is equal to Z_T = 100 / (√2 * cos(45°)) = 100 / (√2 * 0.707) = 100 / √2 ≈ 70.71 ohms.
Therefore, we can write the equation Z_T = R + X_L + X_C as 70.71 = R + 1200Ï€ + 1 / (480Ï€).
Now, trying to solve this equation for R is like trying to solve a riddle in the middle of a clown act. It certainly requires some circus-level math skills! With some numerical juggling, we find that R is approximately equal to 70.71 - 1200Ï€ - 1 / (480Ï€).
So, folks, the value of R is... well, numerically complicated. But remember, life is like a circus – always full of surprises!