JEE MAIN - Physics (2016 - 10th April Morning Slot - No. 23)
Consider a thin metallic sheet perpendicular to the plane of the paper
moving with speed ‘v’ in a uniform magnetic field B going into the plane of the paper (See figure). If charge densities $$\sigma $$1 and $$\sigma $$2 are induced on the left and right surfaces, respectively, of the sheet then (ignore fringe effects) :
_10th_April_Morning_Slot_en_23_1.png)
$$\sigma $$1 = $$ \in $$0 $$\upsilon $$ B, $$\sigma $$2 = $$-$$ $$ \in $$0 $$\upsilon $$ B
$$\sigma $$1 = $${{{ \in _0}\upsilon \,B} \over 2},$$ $$\sigma $$2 = $${{ - { \in _0}\,\upsilon B} \over 2}$$
$$\sigma $$1 = $$\sigma $$2 = $${ \in _0}\,\upsilon B$$
$$\sigma $$1 = $${{ - { \in _0}\upsilon B} \over 2},$$ $$\sigma $$2 = $${{ { \in _0}\upsilon B} \over 2},$$
Explanation
Magnetic force on electron
$$\overrightarrow F $$ = $$-$$ e $$\left( {\overrightarrow V \times \overrightarrow B } \right)$$
F = eVB [As $${\overrightarrow V }$$ and $${\overrightarrow B }$$ are perpendicular]
Also, F = e E
and E = $${\sigma \over {{\varepsilon _0}}}$$
$$ \therefore $$ eVB = e $$ \times $$ $${\sigma \over {{\varepsilon _0}}}$$
$$ \Rightarrow $$ $$\sigma $$ = $${{\varepsilon _0}}$$VB = $$\sigma $$1
as $$\sigma $$1 = $$-$$ $$\sigma $$2
$$ \therefore $$ $$\sigma $$2 = $$-$$ $${{\varepsilon _0}}$$VB
$$\overrightarrow F $$ = $$-$$ e $$\left( {\overrightarrow V \times \overrightarrow B } \right)$$
F = eVB [As $${\overrightarrow V }$$ and $${\overrightarrow B }$$ are perpendicular]
Also, F = e E
and E = $${\sigma \over {{\varepsilon _0}}}$$
$$ \therefore $$ eVB = e $$ \times $$ $${\sigma \over {{\varepsilon _0}}}$$
$$ \Rightarrow $$ $$\sigma $$ = $${{\varepsilon _0}}$$VB = $$\sigma $$1
as $$\sigma $$1 = $$-$$ $$\sigma $$2
$$ \therefore $$ $$\sigma $$2 = $$-$$ $${{\varepsilon _0}}$$VB
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