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An electron gun fires electrons after accelerating them through a potential difference V. Electrons are coming out from a tiny hole. Most of the electrons go straight but some of them make a slightly divergent angle. A uniform magnetic field can be set up along the direction of motion of electrons. These slightly diverged electrons are to be brought to focus at a distance I from the point of exit from the electron gun. If B_(1) , is the minimum magnetic field needed to focus the electrons and B_(2), is the next possible higher value of magnetic field, calculate e/m for the electrons. |
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Answer» Solution :Let `theta` be the angle between velocity of electron and applied magnetic field then we know that electron can be brought into focus if L is integral multiple of pitch of helix. Pitch of the helix can be written as FOLLOWS: ` p = v cos theta xx (2 pi m)/(EB) approx (2 pi m v)/(eB)` Electrons are accelerated through a potential difference V. Thus, we can write the following EXPRESSION for the kinetic energy of the electrons: ` eV = 1//2 mv^(2)` `rArr "" mv = sqrt(2 emV)` Substituting the value of mo, we can rewrite the pitch of the helix as follows: `p = (2 pi sqrt(2 emV))/(eB) = sqrt((8 pi^(2) emV)/(e^(2)B^(2))) = sqrt((8 pi^(2) mV)/(eB^(2))) ` The required magnetic field can be written in terms of pitch of the helix as follows: ` B = 1/p sqrt((8 pi^(2) mV)/e)` Let `e//m = x`,then the above expression can be rewritten as follows: `B = 1/p sqrt((8 pi^(2)V)/x) ` ...(i) For magnetic field `B_(1) `, Lis equal to the pitch and for magnetic field `B_(2)`., L is twice the pitch or we can say the pitch is half the L. We can write equation (i)for the two cases as follows: `B_(1) = 1/L sqrt((8pi^(2)V)/x) ` ...(ii) ` B_(2) = 2/L sqrt((8 pi^(2)V)/x) ` ..(iii) From the above equations, we get : `B_(2) - B_(1) = 1/L sqrt((8 pi^(2)V)/x) ` ` rArr"" (B_(2) - B_(1))^(2) = 1/L^(2) (8 pi^(2)V)/x` `rArr "" x = 1/L^(2) (8 pi^(2)V)/((B_(2) - B_(1))^(2)) ` `rArr "" e/m = ( 8 pi^(2)V)/(L^(2)(B_(2) - B_(1))^(2))` `rArr"" e/m = (8 pi^(2)V)/(L^(2)(B_(2) - B_(1))^(2))` |
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