jangyasinniTeacher

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The current density vector given is ( vec{J} = 20 sin(x) hat{i} + y cos(z) hat{j} ).

To calculate the charge density (( rho )) at a point, we use the continuity equation, which relates the divergence of the current density (( vec{nabla} cdot vec{J} )) to the rate of decrease of charge density over time. The continuity equation in the absence of time-varying charge density is

[ vec{nabla} cdot vec{J} = -frac{partial rho}{partial t} ]

For a stationary charge distribution (( partial rho / partial t = 0 )), the divergence of ( vec{J} ) must be zero for charge conservation:

[ vec{nabla} cdot vec{J} = 0 ]

The divergence (( vec{nabla} cdot vec{J} )) of the current density vector ( vec{J} ) is given by

[ vec{nabla} cdot vec{J} = frac{partial}{partial x}(20 sin(x)) + frac{partial}{partial y}(y cos(z)) + frac{partial}{partial z}(0) ]

[ = 20 cos(x)

b

Explanation: Using continuity equation, the problem can be solved. Div(J) =

– dρ/dt. Div(J) = 20cos x + cos z. At origin, we get 20cos 0 + cos 0 = 21. To get ρ, on

integrating the Div(J) with respect to t, the charge density will be 21t