A battery with EMF 6.0 Volts and internal resistance 0.1 ohms
is connected by a long wire to a resistor R = 8.0 ohms. The
wire goes around a solenoid. The
solenoid has 4000 turns per meter and area 25 cm^2.
a) If the current in the solenoid is I = 200 Amps and is
constant, then what is the current through the 8 ohm resistor?
b) If the current in the solenoid is I(t) = 200 cos(120(pi)t),
then what is the maximum current through the 8 n resis
tor?
c) Is the battery ever being charged when the current in
the solenoid is I(t) = 200cos(120(pi)t)?
To find the current through the 8-ohm resistor, we need to calculate the total current passing through the circuit. Let's break down the problem step by step:
a) Finding the total current:
The total current (I_total) is the sum of the current passing through the resistor (I_resistor) and the current passing through the solenoid (I_solenoid). Since the current is constant (I = 200 Amps), we can assume that the current through the solenoid is equal to I (I_solenoid = 200 Amps).
Using Kirchhoff's voltage law (KVL), we can write the equation:
EMF - (I_total * R_internal) = I_total * R_resistor
Substituting the given values:
6.0 Volts - (I_total * 0.1 ohms) = I_total * 8.0 ohms
Simplifying the equation:
6.0 Volts - 0.1 I_total = 8.0 I_total
Rearranging and solving for I_total:
7.9 I_total = 6.0 Volts
I_total = 6.0 Volts / 7.9
I_total ≈ 0.7595 Amps
The total current passing through the circuit is approximately 0.7595 Amps.
b) Finding the maximum current through the 8-ohm resistor:
Given that the current in the solenoid is represented by the equation I(t) = 200 cos(120πt), we need to find the maximum value of this current.
The maximum value of cos(x) is 1, so the maximum current in the solenoid (I_solenoid_max) is:
I_solenoid_max = 200 Amps * 1
I_solenoid_max = 200 Amps
Since the current passing through the resistor is the same as the current passing through the solenoid, the maximum current through the 8-ohm resistor is also 200 Amps.
c) Determining if the battery is being charged:
To determine if the battery is being charged or not, we need to analyze the direction of the current in the circuit.
Given that the current in the solenoid is represented by the equation I(t) = 200 cos(120πt), we observe that the current oscillates with time due to the cosine function. Therefore, the current alternates between positive and negative values.
If the current in the solenoid during positive half-cycles (when cos(120πt) is positive) is in the same direction as the EMF (6.0 Volts), then the battery is being charged. On the other hand, if the current in the solenoid during positive half-cycles is in the opposite direction of the EMF, then the battery is being discharged.
To determine the direction of the current and if the battery is being charged, we would need more information about the specific orientation and connection of the solenoid and the battery.