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=0.04 A.
Consider a point P on the plane which is equally distant from the two plates and is a distance r=0.0755 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.
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We are looking for the magnetic field B...
Think about where is the point P (inside or outside?)
The formula is written, and is one of these.
Be strong!
To calculate the magnitude of the magnetic field at point P at time t1 during the charge of the capacitor, you can make use of Ampere's Law. Ampere's Law states that the line integral of the magnetic field around a closed loop is equal to the product of the current passing through the loop and the "permeability of free space" constant, μ₀.
Here's the step-by-step process to calculate the magnetic field:
Step 1: Determine the current passing through the loop
In this case, the current passing through the loop is the current with which the capacitor is being charged, which is I=0.04 A.
Step 2: Choose a closed loop passing through point P
In this scenario, a convenient loop to choose is a circular loop with a radius r=0.0755 m. Since point P is equidistant from the two plates, this loop will pass through point P.
Step 3: Calculate the circumference of the loop
The circumference of the circular loop can be calculated using the formula: C = 2πr.
Step 4: Apply Ampere's Law
According to Ampere's Law, the line integral of the magnetic field (B) along the chosen loop is equal to μ₀ times the current passing through the loop. Mathematically, this can be represented as: ∮ B · dl = μ₀I, where ∮ denotes the line integral around the loop and dl represents an element of the loop.
Step 5: Calculate the magnetic field (B)
Since B is constant along the chosen loop (due to symmetry), we can calculate it by dividing the line integral by the circumference of the loop. Mathematically, this can be represented as: B = (∮ B · dl) / C.
In this case, since the loop is a circle, the line integral (∮ B · dl) can be calculated as the product of B and the length of the circumference (C). Therefore, it can be simplified as: (∮ B · dl) = B * C.
Step 6: Substitute the values into the formula
Substitute the known values into the formula: B = (μ₀I) / C.
Step 7: Calculate the magnetic field (B)
Calculate the magnetic field by plugging in the values: B = (μ₀I) / C.
Make sure to convert the units if necessary. The magnetic field will be measured in Tesla (T).
Note: The value of the permeability constant, μ₀, is 4π × 10⁻⁷ T·m/A.
By following these steps, you will be able to calculate the magnitude of the magnetic field at point P at time t1 during the charge of the capacitor.