A long, straight wire of length 2L on the y-axis carries a current I. According to the Biot-Savart Law, the magnitude of the magnetic field due to the current at a point (a,0) is given by

B(a)= (mu sub 0 multiply by I) divided by 4pi times the integral from -L to L of (Sine beta / r^2)dy

where mu sub 0 is a physical constant, a>0, and beta, r, and y are related.

a)Show that the magnitude of the magnetic field at (a,0) is
B(a)= (mu sub 0 times I and L) divided by (2pi*a*sqrt(a^2 + L^2))

b)what is the magnitude of the magnetic field at (a,0) due to an infinitely long wire
(L -> infinity)?

Note: (*) means multiply by

To derive the expression for the magnitude of the magnetic field at point (a, 0), we can follow these steps:

a) Start with the given expression for B(a):

B(a) = (μ₀ * I) / (4π) ∫(-L to L) (sin(β) / r²) dy

b) To simplify the expression, we need to relate β, r, and y. Since the wire is straight and on the y-axis, the distance between the point (a, 0) and the infinitesimally small current element located at (0, y) is given by the Pythagorean theorem:

r² = a² + y²

c) Next, we need to determine β, which represents the angle between the r-vector (from the current element to the point (a, 0)) and the y-axis. Using trigonometry, we find:

sin(β) = a / r

d) Substitute the expressions for sin(β) and r² into the integral:

B(a) = (μ₀ * I) / (4π) ∫(-L to L) [(a / (a² + y²)) / (a² + y²)] dy

e) Now, apply an appropriate substitution to simplify the integral. Let's substitute u = a² + y²:

du = 2y dy

y dy = (1/2) du

f) Update the limits of integration using the substitution. When y = -L, u = a² + (-L)² = a² + L². When y = L, u = a² + L².

B(a) = (μ₀ * I) / (4π) ∫(a² + L² to a² + L²) [(a / u) / u] (1/2) du

g) Simplifying further:

B(a) = (μ₀ * I) / (4π) * (1/2) * ∫(a² + L² to a² + L²) (a / u²) du

B(a) = (μ₀ * I) / (8π) * ∫(a² + L² to a² + L²) (1 / u) du

h) Since the integral is over a constant value, the result is simply the constant multiplied by the difference in the limits of integration:

B(a) = (μ₀ * I) / (8π) * [(1 / (a² + L²)) - (1 / (a² + L²))]

B(a) = (μ₀ * I) / (8π) * [1 - 1]

B(a) = (μ₀ * I) / (8π)

i) Simplifying the expression further:

B(a) = (μ₀ * I) / (2π) * (1 / (8))

B(a) = (μ₀ * I) / (16π)

j) Finally, using the note ( * ) to indicate multiplication:

B(a) = (μ₀ * I * L) / (2π * a * sqrt(a² + L²))

This is the expression for the magnitude of the magnetic field at point (a, 0) due to the long, straight wire of length 2L on the y-axis carrying a current I.

b) Now, let's consider the magnitude of the magnetic field at point (a, 0) due to an infinitely long wire (L → infinity). In this case, we can take the limit of the expression derived above as L approaches infinity:

lim(L→∞) B(a) = (μ₀ * I * L) / (2π * a * sqrt(a² + L²))

As L approaches infinity, the term L/(sqrt(a² + L²)) becomes 1, simplifying the expression to:

B(a) = (μ₀ * I) / (2π * a)

Therefore, the magnitude of the magnetic field at point (a, 0) due to an infinitely long wire is given by (μ₀ * I) / (2π * a).