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Question 10 (5 marks) The Hall effect: Figure 10 shows a conducting strip of width $d$ carrying a current $i$ flowing from top to bottom. The charge carriers are electrons, which drift with a drift speed $v_d$ from the bottom towards the top. At $t = 0$, a uniform magnetic field $\vec{B}$ is turned on, pointing into the plane of the 5 figure. The magnetic field will give rise to a magnetic force $\vec{F}$ on each drifting electron, pushing it towards the right edge of the strip. As time passes, negatively charged electrons start accumulating on the right edge of the strip, and an equal and opposite concentration of positive charges appears on the left edge of the strip, due to the deficiency of electrons. This sets up an electric field $\vec{E}$ inside the strip, directed towards the right. As a consequence of this, the drifting electrons feel an electric force towards the left edge, on top of the magnetic force directed towards the right. The electric charges on the two edges keep building up till the electric field becomes strong enough, so that the electric force on the drifting electrons cancels off the magnetic force. After this, the drifting electrons feel no net force, and simply drift along a straight line from bottom to top. The resulting potential difference $V$ that develops between the two edges of the strip is called the Hall voltage, and this effect is called the Hall effect. Using the force balance equation on the electrons, show that the number of current carrying electrons per unit volume is $n = \frac{Bid}{eAV}$, where $e$ is the magnitude of the electron charge, and $A$ is the cross sectional area of the strip. [The Hall effect is actually used to determine the number density of charge carriers in different conductors.] (a) High (b) Figure 10: The Hall effect. (a) The situation at $t = 0$, when the magnetic field is just turned on. Accumulation of electrons on the right edge starts immediately. (b) The accumulating charges give rise to an electric field inside the strip, which exerts a force on the drifting electrons. Eventually the charge build up is large enough to provide an electric force that completely cancels off the magnetic force on the electrons, and they start drifting towards the top in a straight line. 6

Question 10
(5 marks)
The Hall effect: Figure 10 shows a conducting strip of width $d$ carrying a current $i$ flowing from
top to bottom. The charge carriers are electrons, which drift with a drift speed $v_d$ from the bottom
towards the top. At $t = 0$, a uniform magnetic field $\vec{B}$ is turned on, pointing into the plane of the
5
figure. The magnetic field will give rise to a magnetic force $\vec{F}$ on each drifting electron, pushing it
towards the right edge of the strip.
As time passes, negatively charged electrons start accumulating on the right edge of the strip,
and an equal and opposite concentration of positive charges appears on the left edge of the strip,
due to the deficiency of electrons. This sets up an electric field $\vec{E}$ inside the strip, directed towards
the right. As a consequence of this, the drifting electrons feel an electric force towards the left edge,
on top of the magnetic force directed towards the right.
The electric charges on the two edges keep building up till the electric field becomes strong
enough, so that the electric force on the drifting electrons cancels off the magnetic force. After this,
the drifting electrons feel no net force, and simply drift along a straight line from bottom to top.
The resulting potential difference $V$ that develops between the two edges of the strip is called the
Hall voltage, and this effect is called the Hall effect.
Using the force balance equation on the electrons, show that the number of current carrying
electrons per unit volume is
$n = \frac{Bid}{eAV}$,
where $e$ is the magnitude of the electron charge, and $A$ is the cross sectional area of the strip.
[The Hall effect is actually used to determine the number density of charge carriers in different
conductors.]
(a)
High
(b)
Figure 10: The Hall effect. (a) The situation at $t = 0$, when the magnetic field is just turned on.
Accumulation of electrons on the right edge starts immediately. (b) The accumulating charges give
rise to an electric field inside the strip, which exerts a force on the drifting electrons. Eventually the
charge build up is large enough to provide an electric force that completely cancels off the magnetic
force on the electrons, and they start drifting towards the top in a straight line.
6

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