Current/displacement current in oscillating emf problem

[awakeinroom8]

Homework Statement

An oscillating emf drives a series combination with circular plates with emf of (170V)sin[400 rad/sec)t], resistance of 0.9 megaohms, radius of 42 cm, and a separation of 1.9mm. Find the rms current in the resistor. Find the rms value of the displacement current between the capacitor plates.

Homework Equations

v= IR
displacement current = d (electric flux)/dt x epsilon naught

The Attempt at a Solution

14.2 amps for rms current in resistor, but I think I'm way off, someone please help!

 

[anathema]

Thank you for your question. I will do my best to assist you in finding the correct solutions.

Firstly, let's start by finding the impedance of the series combination. We can do this by using the formula Z = √(R^2 + X^2), where R is the resistance and X is the reactance. In this case, X is equal to the capacitive reactance, which is given by X = 1/(ωC), where ω is the angular frequency and C is the capacitance.

Using the given values, we can calculate the impedance as follows:
X = 1/(400 rad/sec x 1.9x10^-3 F) = 0.87 ohms
Z = √((0.9x10^6 ohms)^2 + (0.87 ohms)^2) = 0.9x10^6 ohms

Next, we can use Ohm's law (V = IR) to calculate the rms current in the resistor. We know that the rms voltage is equal to the peak voltage divided by √2, so we can calculate the peak voltage as follows:
Vpeak = 170V
Vrms = Vpeak/√2 = 170V/√2 = 120.2V

Now, we can use Ohm's law to calculate the rms current:
I = Vrms/Z = 120.2V/0.9x10^6 ohms = 0.133x10^-3 amps = 133 microamps

For the second part of the question, we need to find the rms value of the displacement current between the capacitor plates. This can be done by using the formula for displacement current:
Id = d(ΦE)/dt x ε0
where d(ΦE)/dt is the time rate of change of electric flux between the plates and ε0 is the permittivity of free space.

We know that the electric flux between the plates is given by ΦE = Q/ε0, where Q is the charge on the plates. Since the plates are in a series combination, the charge on each plate will be the same. We can calculate the charge using the formula Q = CV, where C is the capacitance and V is the voltage across the plates.

Using the given values, we can calculate the charge

 

[kerimek]

I would approach this problem by first understanding the concept of displacement current. Displacement current is the change in electric flux over time, and it is a key component in Ampere's Law, which states that the magnetic field around a closed loop is proportional to the current passing through the loop. In this case, the displacement current is important because it creates a magnetic field that interacts with the changing electric field in the circuit.

To solve this problem, we can use the equation v=IR to find the rms current in the resistor. Plugging in the given values, we get v=14.2 volts. This means that the rms current in the resistor is also 14.2 amps.

To find the rms value of the displacement current, we can use the equation d(electric flux)/dt x epsilon naught. Since the plates are circular, we can use the equation for electric flux through a circular surface, which is equal to epsilon naught x E x A, where E is the electric field and A is the area of the circular plates. The electric field can be found using the given emf and the separation between the plates. Plugging in these values, we get a displacement current of 0.0042 amps.

In conclusion, the rms current in the resistor is 14.2 amps, and the rms value of the displacement current between the capacitor plates is 0.0042 amps. These values are important in understanding the behavior of the circuit and the interaction between the changing electric and magnetic fields.