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.
I agree with the hint!
What is the question here?
Here is an MIT solution..
http://ocw.mit.edu/courses/physics/8-02-electricity-and-magnetism-spring-2002/assignments/pss5.pdf
see prob 5.3
To find the magnetic field at point P above the center of the strip, we need to use the Biot-Savart Law. The Biot-Savart Law states that the magnetic field (dB) at a point due to a small segment of current-carrying wire is directly proportional to the current (I), the length element (dl), the sine of the angle between the line connecting the segment to the point (θ), and inversely proportional to the square of the distance from the segment to the point (r^2).
The differential equation for the magnetic field is given by:
dB = (μ₀ / 4π) * (I * dl × r̂) / r^2
Where:
- μ₀ is the permeability of free space (4π × 10^(-7) Tm/A)
- I is the current through the strip (10 A)
- dl is the length element along the strip
- r is the distance from the length element to the point P
To find the magnetic field at point P, we need to integrate the contributions from all the length elements along the strip.
Since the strip is infinitely long, we can consider an element of width (dx) along the strip. The length of this element (dl) is given by dl = √(dx^2 + a^2).
Now, let's set up the integral to find the magnetic field at point P.
B = ∫ (μ₀ / 4π) * (I * dl × r̂) / r^2
To simplify the expression, we can utilize the fact that the sin(θ) term and the vector product dl × r̂ are perpendicular to each other. Therefore, the vector product reduces to the magnitude of dl multiplied by the magnitude of r̂. This simplification allows us to rewrite the equation as:
B = ∫ (μ₀ / 4π) * (I * dl) * (dl × r̂) / r^2
We can rewrite the cross product (dl × r̂) as dl * r̂ * sin(θ). Since sin(θ) is the same for all elements along the strip, we can take it out of the integral:
B = (μ₀ * I / 4π) * sin(θ) * ∫ (dl * r̂) / r^2
Now, we substitute dl = √(dx^2 + a^2) into the equation and rewrite r^2 as (x^2 + a^2) using the Pythagorean theorem:
B = (μ₀ * I / 4π) * sin(θ) * ∫ (√(dx^2 + a^2) * r̂) / (x^2 + a^2)
Finally, we integrate this expression from -∞ to +∞ since the strip extends infinitely in both directions:
B = (μ₀ * I / 4π) * sin(θ) * ∫[from -∞ to +∞] (√(dx^2 + a^2) * r̂) / (x^2 + a^2)
To solve the integral, a substitution is usually made, but the calculation becomes complex. In this case, it is best to use a software or calculator that can perform symbolic integration.
Once you obtain the symbolic solution, you can plug in the values for sin(θ = 90°), I = 10 A, μ₀ = 4π × 10^(-7) Tm/A, and the desired values for a and w to determine the magnetic field at point P.