Consider a thin, infinitely long conducting ribbon that carries a uniform current density j(current per unit area). The width of the ribbon is w and its thickness s is extremely small (s<<w).P is a point in the plane of the ribbon, at a large distance (x>>s) from the ribbon edge.

Question

Consider a thin, infinitely long conducting ribbon that carries a uniform current density j(current per unit area). The width of the ribbon is w and its thickness s is extremely small (s<<w).P is a point in the plane of the ribbon, at a large distance (x>>s) from the ribbon edge.

What is the magnitude of the magnetic field B (in T ) at point P for the following values of w , j , s and x ?
w= 6 cm ; s= 0.1 cm; j= 1A/m^2 and x=21 cm .

Please can someone give me a hand to start this exercice??

Answer 1

I have solved it, but i don´t know if the result is correct. The result that i obtain is:

B= 1.39*10^-10[T]

can someone comfirm it

Thanks

Answer 2

To solve this exercise, we can use the Biot-Savart Law, which relates the magnetic field produced by a current-carrying wire to the current density.

The Biot-Savart Law states that the magnetic field at a point P due to a small segment of the wire is given by:

d𝐁 = (𝜇₀/4𝜋) * (𝐽 * d𝐥 x 𝐫) / r²

Where:
- d𝐁 is the magnetic field at point P due to a small segment of the wire
- 𝜇₀ is the permeability of free space (4𝜋 x 10⁻⁷ T·m/A)
- 𝐽 is the current density (current per unit area)
- d𝐥 is the infinitesimal length element of the wire
- 𝐫 is the position vector connecting the current element to the point P
- r is the distance between the current element and the point P

Given that we have a conducting ribbon instead of a wire, we can consider that the ribbon is made up of an infinite number of small wire segments. So we can integrate the Biot-Savart Law over the entire ribbon to find the total magnetic field at point P.

Let's break down the steps to solve the exercise:

1. Find the current passing through a small section of the ribbon.
- The current is given by the current density times the area of the small section.
- In this case, the current passing through the small section is j * (w * s).

2. Calculate the infinitesimal magnetic field produced by a small section of the ribbon using the Biot-Savart Law.
- Use the formula: d𝐁 = (𝜇₀/4𝜋) * (current * d𝐥 x 𝐫) / r².
- The current is obtained from step 1.
- The infinitesimal length element d𝐥 is taken along the width of the ribbon (since it is infinitely long).
- 𝐫 is the position vector from the current element to point P.

3. Integrate the magnetic field over the entire length of the ribbon (along the width direction).
- Integrate the expression obtained in step 2 with respect to the width coordinate, from 0 to w.
- This integration gives the total magnetic field at point P due to the entire ribbon.

4. Substitute the given values (w, s, j, x) into the integrated expression to calculate the magnitude of the magnetic field at point P.

Now, you can start by finding the current passing through a small section of the ribbon using the current density and the area of the section.