Question

An electron beam has an aperture $1.0m{m^2}$. A total of $6 \times {10^6}$ electrons go through any perpendicular cross section per second. Find the current density in the beam. (in $\dfrac{A}{m^2}$)
A) $9.1 \times {10^{13}}$
B) $9.6 \times {10^3}$
C) $6.6 \times {10^5}$
D) $8.6 \times {10^{11}}$

Answer 1

Hint:The question is from electricity. Substitute the values in the current density equation and find the value of current density (J).


Formula Used:
\[J = \dfrac{I}{A}\]
Where J = current density, I = current and A = cross-section area.
$I = \dfrac{Q}{t}$
Where I = charge, Q = charge and t = time.
$Q = ne$
Where, Q = charge, n = number of electrons and e = charge of an electron.

Complete answer:
The charge Q is given by,
$Q = ne = (6.0 \times {10^{16}})(1.6 \times {10^{ - 19}})$
$Q = 9.6 \times {10^{ - 3}}C$
Substitute the value of charge and find the current I.
$I = \dfrac{Q}{t} = \dfrac{{9.6 \times {{10}^{ - 3}}}}{1}$
$I = 9.6 \times {10^{ - 3}}A$
Substitute the value of current and find current density J.
$J = \dfrac{I}{A} = \dfrac{{9.6 \times {{10}^{ - 3}}}}{{1.0 \times {{10}^6}}}$
$J = 9.6 \times {10^3}A{m^{ - 2}}$

Hence, the correct option is Option (B) $9.6 \times {10^3}$.

Additional Information:
Drift velocity is the average velocity attained by charged particles, (electrons) in a material due to an electric field (\[\overrightarrow E \]). The SI unit of drift velocity is the same as velocity which is m/s. The drift velocity and current flowing through the conductor both increase as the intensity of the electric field increases.
The relation between drift velocity and current is given below.
\[{v_d} = \dfrac{I}{{neA}}\]
\[{v_d}\]= drift velocity
I = current flow
n = free electron density
e = charge of an electron
A = cross sectional area
Mobility (\[\mu \]) of an electron is the drift velocity of an electron for a unit electric field (\[E\]). The equation for mobility is given below.
\[\mu = \dfrac{{{v_d}}}{E}\]


Note: The relation between drift velocity and Current Density is \[J = n{v_d}e\]. Where, n = free electron density, e = charge of an electron, \[{v_d}\]= drift velocity and J = current density. In quantization of charges n is defined as no. of electrons getting transferred but when we are using drift velocity and current density density formula there n is defined as no. of electrons per unit volume. So always pay attention while putting these values.