Current Through a Cylindrical Region in a Static Magnetic Field

Problem Statement

In a three-dimensional Cartesian coordinate system $(x, y, z)$(x,y,z) in vacuum, a steady magnetic flux density $\mathbf{B}(x, y, z)$B(x,y,z) is distributed throughout space as follows:

$$\mathbf{B}(x, y, z) = B_0 e^{-\alpha(x^2 + y^2)} (-y \mathbf{i} + x \mathbf{j})$$B(x,y,z)=B0​e−α(x2+y2)(−yi+xj)

where $B_0$B0​ and $\alpha$α are positive constants, and $\mathbf{i}, \mathbf{j}$i,j are unit vectors along the positive $x$x- and $y$y-axes, respectively.

A steady current flows through space, governed by Maxwell's equations, to produce this magnetic field distribution.

Find the total current $I$I passing in the $+z$+z direction through the disk region $S$S centered at the origin of the $xy$xy-plane with radius $R = \frac{1}{\sqrt{\alpha}}$R=α​1​ (defined by $x^2 + y^2 \le R^2$x2+y2≤R2, $z = 0$z=0).

The permeability of vacuum is $\mu_0$μ0​ and $e$e denotes Napier's number (Euler's number).

Constraints

• Magnetic field coefficient: $B_0 = 1.37 \times 10^{-2} \text{ T}$B0​=1.37×10−2 T
• Spatial decay coefficient: $\alpha = 2.50 \times 10^3 \text{ m}^{-2}$α=2.50×103 m−2
• Permeability of vacuum: $\mu_0 = 4\pi \times 10^{-7} \text{ N/A}^2$μ0​=4π×10−7 N/A2

Input Format

The current $I$I is expressed using Napier's number $e$e as $I = \frac{X}{e} \text{ (A)}$I=eX​ (A), where $X$X is an irreducible fraction $\frac{A}{B}$BA​ with $A$A and $B$B being coprime positive integers. Give the value of $A + B$A+B as a positive integer.