A straight wire piece of length x carries a current i . A point P is located on the perpendicular bisector of this piece, at a distance d from the middle point. How does the magnetic field at point P vary with distance d in the limits d x and d x , respectively?

Step-by-step solution

The magnetic field at a point on the perpendicular bisector of a finite straight wire of length x carrying current i at a distance d is given by Biot-Savart law: B = ₀ i 4 d ( ₁ + ₂) Since the point is on the perpendicular bisector, ₁ = ₂ = . From the geometry of the figure, = x/2 d^2 + x^2 4 . Substituting this into the magnetic field expression: B = ₀ i 4 d ( x d^2 + x^2 4 ) For the limit d x , the term x^2 4 can be neglected compared to d^2 . Thus, d^2 + x^2 4 d . B ₀ i x 4 d^2 Therefore, B 1 d^2 for d x . For t