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).