# . Find the magnitude of the magnetic field at a distance (a) r=0, (b) r = 3.4 mm and (c) r=5.0 mm from the center.

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Question

# The current density inside a long, solid, cylindrical wire of radius a = 5.0 mm is in the direction of the central axis and its magnitude varies linearly with radial distance r from the axis according to J = J0r/a, where J0 = 290 A/m2. Find the magnitude of the magnetic field at a distance (a) r=0, (b) r = 3.4 mm and (c) r=5.0 mm from the center.

The current density inside a long, solid, cylindrical wire of radius a = 5.0 mm is in the direction of the central axis and its magnitude varies linearly with radial distance r from the axis according to J = J0r/a, where J0 = 290 A/m2. Find the magnitude of the magnetic field at a distance (a) r=0, (b) r = 3.4 mm and (c) r=5.0 mm from the center.

$j = j_0\frac {r}{a}$

(a)

At $r = 0$, $B = 0$

(b)

$I =\int_{0}^{R} \vec {j} . \vec {ds} = \frac {j_0}{a}\int_{0}^{R} 2\pi r ^2 dr= \frac {2\pi j_0 R^3}{3a}$

$R = 3.4 \: mm = 3.4 \times 10^{-3}\: m$

$I = 4.77 \: mA$

$B \times 2\pi R = \mu_0 I$ (Ampere's law)

$B = \frac {\mu_0 I}{2\pi R} = 2.80 \times 10^{-7} \: T$

(c)

$I =\int_{0}^{R} \vec {j} . \vec {ds} = \frac {j_0}{a}\int_{0}^{R} 2\pi r ^2 dr= \frac {2\pi j_0 R^3}{3a}$

$R = 5 \: mm = 5 \times 10^{-3}\: m$

$B = \frac {\mu_0 j_0 R^2}{3a} = 6.07 \times 10^{-7} \: T$