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CBSE Class 12-science Answered

A semiconductor slab of thickness t=0.5 mm with length l=30 cm carrying current I=2A placed in a magnetic field B=0.25T directed into the page, perpendicular to the flat surface of the slab carries 7*10^24 electrons per m^3. Calculate the Hall voltage across the slab. Indicate which edge (top or bottom) is at higher potential and why?

Asked by arjunsah797 | 26 Jan, 2022, 02:56: PM
Expert Answer
Figure shows a semiconductor slab of length l , width w and thickness t . 
 
Let us assume cartesian coordinate system so that length of slab is along y-direction and width is along z-direction as shown in figure.
 
Let Current I is passing through the semiconductor slab along y-direction .
 
Let magnetic field B is directed into the page, i.e. along -x-axis direction.
 
Since current is passing along y-axis direction, drift velocity vd of electron is along -y-axis direction as shown in figure.
 
Force acting on electron F = -e ( vd × B )
 
where e is magnitude of charge on electron
 
Direction of force is along z-direction . Hence electrons accumulate at top of slab and gives -ve potential  ,
while bottom of slab gets +ve potential as shown in figure.
 
Hall voltage,  VH = ( vd × B × w )  ...............................(1)
 
Current and drift velocity are related as, I = ne × e  × vd × A  ............................(2)
 
where ne is electron density  and A = ( w × t ) is area of cross section
 
Substituting vd from eqn.(2) , we get Hall voltage from eqn.(1) as
 
begin mathsize 14px style V subscript H space equals space fraction numerator B cross times I over denominator n subscript e space cross times e space cross times space t end fraction space end style
begin mathsize 14px style V subscript H space equals space fraction numerator 0.25 space cross times space 2 over denominator 7 cross times 10 to the power of 24 cross times 1.602 space cross times 10 to the power of negative 19 end exponent cross times 0.5 space cross times 10 to the power of negative 3 end exponent end fraction space equals space 8.917 space cross times space 10 to the power of negative 3 end exponent space V space equals space 8.917 space m V end style
Answered by Thiyagarajan K | 26 Jan, 2022, 08:15: PM
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