Consider the process of charging a parallel plate capacitor with circular plates of radius R=5 cm separated by a distance d=0.2 cm. At some time t1, the capacitor is being charged with a current I=0.075 A.

Consider a point P on the plane which is equally distant from the two plates and is a distance r=0.0505 m away from the axis of the capacitor. (a) Calculate the magnitude of the magnetic field (in Tesla) at a point P at time t1 during the charge of the capacitor.

(b) Calculate the magnitude of the magnetic field (in Tesla) at a point P after the capacitor is fully charged.

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To calculate the magnetic field at point P, we can use Ampere's law, which states that the line integral of the magnetic field around a closed loop is equal to the product of the current enclosed by the loop and the permeability of free space.

a) At time t1, when the capacitor is being charged with a current I=0.075 A:

1. Firstly, we need to find the current passing through the loop defined by point P. Since the capacitor is being charged, we know that the current is continuously increasing. Therefore, the current passing through point P is given by the rate of change of charge with respect to time.

I = dQ/dt

2. The charge on a capacitor is given by Q = C * V, where C is the capacitance and V is the voltage across the capacitor plates. Since the capacitor is still being charged, the voltage will be changing. We can express the voltage as an increasing function of time:

V = V0 * (1 - e^(-t/RC))

Here, V0 represents the final voltage, R is the resistance in the circuit, and C is the capacitance.

3. We can relate the voltage across the capacitor to the current passing through it using Ohm's law:

V = IR

4. By substituting V from step 2 into the above equation and solving for I, we can find the current as a function of time:

I = (V0/R) * (1 - e^(-t/RC))

5. Now, substitute the given values for I, R, and C into the equation from step 4 and calculate the current passing through point P at time t1.

6. Once we have the current passing through point P, we can apply Ampere's law. We need to choose a closed loop that encloses the current passing through point P. In this case, we can choose a circular loop centered on the axis of the capacitor with a radius r=0.0505 m.

7. Calculate the magnetic field B by evaluating the line integral of the magnetic field along the chosen loop:

B * 2πr = μ0 * I_enclosed

Here, μ0 is the permeability of free space, and I_enclosed is the current passing through the chosen loop.

Solve for B to get the magnitude of the magnetic field at point P at time t1.

b) After the capacitor is fully charged, the current passing through the loop becomes zero. Therefore, the magnetic field at point P will be zero.

Note: This explanation outlines the general steps to solve the problem. Make sure to substitute the given values and the appropriate formulas to get the numerical answers.