A long, straight, cylindrical wire of radius R carries a current uniformly distributed over its cross section.
1. At what location is the magnetic field produced by this current equal to quarter of its largest value? Consider points inside the wire.
2.At what location is the magnetic field produced by this current equal to quarter of its largest value? Consider points outside the wire.
What is the difference between questions 1 and 2?
To find the locations where the magnetic field produced by the current in the wire is equal to a quarter of its largest value, we can use the Biot-Savart Law.
1. Inside the wire:
Consider a current element dI on the wire at a distance x from the center of the wire.
The magnetic field produced by this current element at a point P inside the wire is given by the Biot-Savart Law:
dB = (μ₀/4π) * (dI * sin(θ))/r²
Here, μ₀ is the permeability of free space, dI is the current element, θ is the angle between the current element and the line connecting the current element to the point P, and r is the distance between the current element and the point P.
To calculate the total magnetic field at point P due to the entire wire, we integrate the magnetic field dB over the entire length L of the wire:
B = ∫ (μ₀/4π) * (dI * sin(θ))/r²
Since the wire carries the current uniformly distributed over its cross-section, the current element can be expressed as dI = I₀ * (2πr * dr)/(πR²), where I₀ is the current of the wire, r is the distance from the center of the wire, and dr represents an infinitesimal length element along the wire.
Substituting this expression for dI, we can rewrite the magnetic field equation as:
B = ∫ (μ₀/4π) * [(I₀ * (2πr * dr)/(πR²) * sin(θ))/r²]
To simplify further, we can cancel out terms and integrate:
B = (μ₀ * I₀)/(2R) * ∫ (sin(θ))/r * dr
Integrating this equation gives us the magnetic field B at point P inside the wire. To find the location where the magnetic field is equal to a quarter of its largest value, we need to solve the equation B = (1/4)B_max.
2. Outside the wire:
Outside the wire, the magnetic field produced by the current can be approximated as that of a long straight wire. The magnetic field at a point P outside the wire is given by:
B = (μ₀ * I₀)/(2πr)
To find the location where the magnetic field is equal to a quarter of its largest value, we need to solve the equation B = (1/4)B_max.
Keep in mind that the exact solution may require more advanced mathematical techniques depending on the specific values of R, I₀, and the desired degree of accuracy.