An infinitely long thin metal strip of width w=12cm carries a current of I=10A that is uniformly distributed across its cross section. What is the magnetic field at point P a distance a=3cm above the center of the strip?
I have tried using the integration method but cannot get the correct answer :(
Use symettry as you integrate across. Starting from one side, all you want to add is the horizontal component (the vertical component will be oposite direction when you get to the other side).
So integrate the cosine/sine Theta part of the angle only (theta equals an angle from the strip to the point, the cos/sin will depend on how you label the angle).
I don't understand.
To find the magnetic field at point P above the center of the strip, you can use Ampere's Law.
Ampere's Law states that the magnetic field around a closed loop is directly proportional to the current passing through the loop. In mathematical form, it can be written as:
∮B · dl = μ0 * Ienc
Where:
- ∮B · dl is the line integral of the magnetic field around the closed loop
- μ0 is the permeability of free space (μ0 = 4π × 10^-7 T·m/A)
- Ienc is the enclosed current within the loop
In this case, we can consider a rectangular closed loop that encloses the current-carrying strip. The width of the loop would be equal to the width of the strip (w), and the length of the loop can be chosen as any distance.
Let's assume that the length of the loop is L, and it is perpendicular to the strip width. Thus, two sides of the loop will be horizontal and parallel to the strip, and the other two sides will be vertical. The magnetic field will be in the tangential direction (circumferential) around the loop.
Now, we need to calculate the magnetic field at point P, which is located a distance a above the center of the strip. Since point P is above the center, we can reflect the loop to the bottom half, forming a rectangular loop symmetrically around the strip.
To calculate the magnetic field at point P, we need to find the enclosed current within the loop. In this case, the enclosed current is half of the total current, as we consider only the bottom half of the strip.
Ienc = I/2 = 10A / 2 = 5A
Now, we can apply Ampere's Law to find the magnetic field at point P using the relation:
∮B · dl = μ0 * Ienc
Since the sides of the loop that are parallel to the strip do not contribute to the magnetic field, we are left with the vertical sides for the line integral. The integral can be simplified as follows:
B * L + B * L + B * w + B * w = μ0 * Ienc
2B * L + 2B * w = μ0 * Ienc
2B * (L + w) = μ0 * Ienc
Simplifying further:
B = (μ0 * Ienc) / (2 * (L + w))
Substituting the given values:
μ0 = 4π × 10^-7 T·m/A
Ienc = 5A
L = 2a (since a is the distance above the center on one side)
w = 12cm = 0.12m
B = (4π × 10^-7 T·m/A * 5A) / (2 * (2a + 0.12m))
Finally, substitute the value of a = 3cm = 0.03m:
B = (4π × 10^-7 T·m/A * 5A) / (2 * (2 * 0.03m + 0.12m))
Simplify the expression and calculate it to find the value of the magnetic field at point P.