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 as shown? This one requires some integration! I suggest you solve it symbolically first, then plug in numbers.
So what is your question? This is an integration problem. Draw the figure, and do the integration. Use symettry.
To find the magnetic field at point P, we can use Ampere's Law. Ampere's Law relates the magnetic field around a closed loop to the current passing through that loop.
However, in this case, we need to consider only a small segment of the strip, considering it as an infinitely long straight wire. The current passing through this segment will create a magnetic field at point P.
Let's consider a small element of length "dl" on the strip at a distance "x" from the center of the strip. The current passing through this small segment is given by "dI".
Now, we can apply Ampere's Law to find the magnetic field at point P.
Step 1: Determine an expression for the "dI".
Since the current is uniformly distributed across the cross-section of the strip, the current passing through the small segment "dl" is given by:
dI = (I/w) * dl
Step 2: Determine the expression for the magnetic field "dB" created by "dI" at point P.
Using Ampere's Law, the magnetic field "dB" generated by "dI" is given by:
dB = (μ0/4π) * (dI/r)
Where:
μ0 is the permeability of free space (μ0 = 4π × 10^-7 T·m/A),
r is the distance from the small segment "dl" to point P.
Step 3: Integrate the magnetic field contributions "dB" over the entire length of the wire.
To find the total magnetic field at point P, we need to integrate the magnetic field contributions (dB) along the entire length of the wire.
dB_total = ∫dB
Integrating over the entire length of the wire, the integration limits will go from -∞ to +∞.
Step 4: Simplify the integral and solve for the magnetic field "B".
Once we have the integral expression for dB_total, we can simplify it and solve for the magnetic field "B".
B = ∫dB
Now, you can plug in the values of the parameters (I, w, a) into the expression for "B" and evaluate it numerically.