The output voltage of an AC generator is given by Δv = (180 V) sin (70πt). The generator is connected across a 13.0-Ω resistor.

(a) By inspection, what is the maximum voltage?
V

(b) By inspection, what is the frequency?
Hz

(c) Find the rms voltage across the resistor.
V

(d) Find the rms current in the resistor.
A

(e) Find the maximum current in the resistor.
A

(f) Find the power delivered to the resistor.
W

(g) Find the current when t = 0.0050 s.
A

U = 180•sin (70πt).

Umax = 180 V
f=ω/2π=70π/2π =35 Hz,
rms U=Umax/√2= =0.707•Umax=0.707•180=127.3 V
U/R = (180/R)•sin (70πt)
I=(180/13) •sin (70πt)=13.85•sin (70πt)
Imax =13.85 A
rmsI= Imax/√2=0.707•Imax=
=0.707•13.85=9.8 A
Pmax=Imax•Umax=180•13.85=2493 W
P=I•U= 127.3•9.8 =1244.6W

I=13.85•sin (70πt)=
=13.85•si(70π •0.005)=
=13.85•0.89 =12.34 A

(a) By inspection, what is the maximum voltage?

The maximum voltage can be determined by looking at the coefficient in front of the sine function, which is 180 V.

(b) By inspection, what is the frequency?
Well, it seems that we have a 70πt term in there, so the frequency would be determined by the coefficient of t. However, since we do not have it, let's calculate that: 70πt = 2πf. Solving for f, we get f = 35 Hz. That's quite shocking!

(c) Find the rms voltage across the resistor.
To find the rms voltage, we need to remember that it is equal to the maximum voltage divided by the square root of 2. So, V = 180 V / √2.

(d) Find the rms current in the resistor.
Since we already have the rms voltage, let's calculate the rms current using Ohm's Law, V = I * R. Rearranging, we get I = V / R.

(e) Find the maximum current in the resistor.
To find the maximum current, we need to remember that it is equal to the maximum voltage divided by the resistor's value. Therefore, I = 180 V / 13.0 Ω.

(f) Find the power delivered to the resistor.
The power delivered to the resistor can be calculated using the formula P = V^2 / R.

(g) Find the current when t = 0.0050 s.
Well, to find the current at a specific time, we need to substitute that value into our equation. So, plug in t = 0.0050 s into the equation Δv = (180 V) sin (70πt) and solve for Δv. Then we can use Ohm's Law to find the current, I = Δv / R.

I apologize if my jokes generate more resistance than assistance in this case.

(a) By inspection, the maximum voltage is the amplitude of the sine function. Therefore, the maximum voltage is 180 V.

(b) By inspection, the coefficient of "t" in the sine function represents the angular frequency (ω) of the generator. In this case, ω = 70π. To find the frequency (f), we can use the formula f = ω / (2π). Thus, the frequency is 35 Hz.

(c) The rms (root mean square) voltage can be obtained by multiplying the maximum voltage by 1 / √2. Therefore, the rms voltage across the resistor is (180 V)(1 / √2) ≈ 127.28 V.

(d) The rms current can be found using Ohm's law, V = IR. Rearranging the equation, we have I = V / R. Substituting the values, I = 127.28 V / 13.0 Ω ≈ 9.79 A.

(e) The maximum current can be calculated by dividing the maximum voltage by the resistance, using Ohm's law. Hence, the maximum current is 180 V / 13.0 Ω ≈ 13.85 A.

(f) The power delivered to the resistor is given by the formula P = V^2 / R. Plugging in the values, we have P = (127.28 V)^2 / 13.0 Ω ≈ 1243.91 W.

(g) To find the current when t = 0.0050 s, we substitute this value into the given equation for Δv and divide by the resistance. Therefore, the current at t = 0.0050 s is (∆v / R) = (180 V)sin(70π(0.0050 s)) / 13.0 Ω ≈ 8.48 A.

(a) To find the maximum voltage, we need to look at the amplitude of the function. In this case, the maximum voltage is given by the coefficient of the sine function, which is 180 V.

(b) The frequency of the AC signal can be determined by looking at the coefficient of 't' inside the sine function. In this case, it is 70π. We can divide this value by 2π to get the frequency in hertz (Hz):
Frequency = (70π)/(2π) = 35 Hz.

(c) The formula for calculating the rms voltage in AC circuits is Vrms = Vmax / √2. We can substitute the maximum voltage we found in part (a) into this formula to find the rms voltage across the resistor:
Vrms = 180 V / √2 = 127.3 V.

(d) To find the rms current in the resistor, we need to use Ohm's Law, which states that V = I * R, where V is the voltage, I is the current, and R is the resistance. Rearranging this equation, we get I = V / R. Substituting the rms voltage from part (c) and the resistance of 13.0 Ω, we can find the rms current:
I = 127.3 V / 13.0 Ω = 9.8 A (rounded to one decimal place).

(e) The maximum current can be calculated as the maximum voltage divided by the resistance:
Imax = Vmax / R = 180 V / 13.0 Ω = 13.8 A (rounded to one decimal place).

(f) Power can be calculated using the formula P = Vrms * Irms, where P is power, Vrms is the rms voltage, and Irms is the rms current. Substituting the values we found in parts (c) and (d), we get:
P = 127.3 V * 9.8 A = 1249.5 W (rounded to one decimal place).

(g) To find the current when t = 0.0050 s, we can substitute this value into the given equation for the output voltage and solve for the current. Substituting t = 0.0050 s into Δv = (180 V) sin (70πt):
Δv = (180 V) sin (70π * 0.0050 s) = (180 V) sin (0.35π)
Using a calculator to find the sine of 0.35π, we get approximately 0.999, so:
Δv = (180 V) * 0.999 = 179.82 V (rounded to two decimal places).

Since the resistor is connected across the generator, the voltage across the resistor is the same as the output voltage, so the current can be calculated as:
I = Δv / R = 179.82 V / 13.0 Ω = 13.8 A (rounded to one decimal place).

Therefore, the current when t = 0.0050 s is 13.8 A, the same as the maximum current.