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A non-uniform magnetic field \( \mathbf{B}=B_{0}\left(1+\frac{y}{d}\right)(-\hat{\mathbf{k}}) \) is present in region of space in between \( y=0 \)
\( \mathrm{P} \) and \( y=d \). The lines are shown in the figure. A particle of mass \( m \) and positive charge \( q \) is moving. Given, an initial velocity \( \mathbf{v}=v_{0} \hat{\mathbf{i}} \). The \( x \)-component of velocity of the particle is
W \( v_{x}=v_{0}-\frac{k q B_{0} d}{2 m} \) when it leaves the field, then find the value of \( k \).
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