Question

From the Biot-Savart law, which of the following is most useful for estimating the magnetic field from a short current element of length L with current i? From the Biot-Savart law, which of the following is most useful for estimating the magnetic field from a short current element of length L with current i? μ0i/2πL μ0i2L μ0iL/4πr2 μ0i/2L

Answer 1

B = μ₀ I / 2π L

Step-by-step explanation:

The biot-Savart law has the equation

B = μ₀ I / 4π ∫ ds * r ^ / r²

Where the vector product gives the direction of the field, ds is taken along the wire length

ds * r ^ = ds sin θ

B = μ₀ I / 4π I ds sinθ / r²

r is the distance from current element to the point of interest, to make the calculation easier suppose that the wire is on the x-axis and the calculation point on the Y-axis at a distance L

ds = dx

We have to write the integral in terms of a single integration variable, if we use geometry, we can write everything in function angle (θ)

sin θ = L / r

r = L / sin θ = L csc θ

tan θ = L / x

x = L / tan θ

dx = L csc²θ dθ

substitutions and calculate

B = μ₀ I / 4π I (L csc²θ dθ) sinθ / (L cscθ)²

B = μ₀ I / 4π I dθ sinθ / a

B = μ₀ I / 4π I dθ sinθ / a

B = μ₀ I 4π L (cos T1-cosT2)

If the wire is very long θ1 = 0 t θ2 =π (cos θ1 -cosθ2) = 2

B = μ₀ I / 2π L

This is the solution