A uniform sheet of surface current density Ks=Ky ay in z=0 plane. find the magnetic field strength H everywhere using ampere's law.
To find the magnetic field strength, H, everywhere using Ampere's Law, you need to follow these steps:
Step 1: Define a Closed Loop
Start by choosing a closed loop that encloses the current-carrying sheet. In this case, you can choose a rectangular loop in the z = 0 plane.
Step 2: Determine the Current Passing Through the Loop
Next, calculate the total current passing through the loop. In this case, the sheet has a uniform surface current density given by Ks = Ky ay. Since the surface current is parallel to the loop in the z = 0 plane, the current passing through the loop is equal to the current density multiplied by the width of the loop. Therefore, the current, I, passing through the loop is given by:
I = Ks * w,
where w is the width of the loop.
Step 3: Choose an Amperian Path
Now, inside the loop you have chosen, select an Amperian path. This path should be a closed curve that lies on the loop you initially chose.
Step 4: Apply Ampere's Law
Apply Ampere's Law to determine the magnetic field strength, H, everywhere on the Amperian path. Ampere's Law states that the line integral of the magnetic field, H, around a closed path is equal to the total current passing through the path. Mathematically, it can be written as:
∮H · dl = I,
where ∮ denotes a closed line integral, H is the magnetic field strength, dl is the differential vector element along the path, and I is the total current passing through the path.
Step 5: Evaluate the Integral
Evaluate the line integral on the left-hand side of Ampere's Law equation. Since H is constant along the path defined by the Amperian path, you can take it out of the integral. The line integral simply becomes:
H ∮dl = H * Circumference of the Amperian path.
Step 6: Substitute Values and Solve for H
Substitute the values of I (from Step 2) and the circumference of the Amperian path into the Ampere's Law equation. Then solve for H:
H * Circumference of the Amperian path = I.
Finally, solve for H:
H = I / Circumference of the Amperian path.
By following these steps, you can determine the magnetic field strength, H, everywhere using Ampere's Law for a uniform sheet of surface current density Ks = Ky ay in the z = 0 plane.