Question

An electron initially 3.00 m from a nonconducting infinite sheet of uniformly distributed charge is fired toward the sheet. The electron has an initial speed of 420 m/s and travels along a line perpendicular to the sheet. When the electron has traveled 2.00 m, its velocity is instantaneously zero, and it then reverses its direction. What is the surface charge density on the sheet?

Answer 1

Answer:

$4.4443704375\times 10^{-18}\ C/m^2$

Explanation:

$\epsilon_0$ = Permittivity of free space = $8.85\times 10^{-12}\ F/m$

$\Delta l$ = Distance charge traveled = 2 m

v = Velocity of electron = 420 m/s

E = Electric field

$m_e$ = Mass of electron = $9.11\times 10^{-31}\ kg$

$q_e$ = Charge of electron = $1.6\times 10^{-19}\ C$

As the energy of the system is conserved we have

$q_eE\Delta l=\dfrac{1}{2}m_ev^2\\\Rightarrow E=\dfrac{1}{2}\dfrac{m_e}{q_e}\times \dfrac{v^2}{\Delta l}\\\Rightarrow E=\dfrac{1}{2}\dfrac{9.11\times 10^{-31}}{1.6\times 10^{-19}}\times \dfrac{420^2}{2}\\\Rightarrow E=2.51094375\times 10^{-7}\ N/C$

For an infinite non conducting sheet electric field is given by

$E=\dfrac{\sigma}{2\epsilon}\\\Rightarrow \sigma=2E\epsilon\\\Rightarrow \sigma=2\times 2.51094375\times 10^{-7}\times 8.85\times 10^{-12}\\\Rightarrow \sigma=4.4443704375\times 10^{-18}\ C/m^2$

The surface charge density is $4.4443704375\times 10^{-18}\ C/m^2$

Answer 2

The surface charge density on the sheet is 1.77 x 10⁻¹⁷ C/m².

The given parameters;

• initial position of the electron, x₁ = 3 m
• initial speed of the electron, v = 420 m/s
• final position of the electron, x₂ = 2m

The electric field on the charged electron is calculated from the principle of conservation of energy;

$F\Delta d= \frac{1}{2} mv^2\\\\Eq \Delta d = \frac{1}{2} mv^2\\\\E = \frac{mv^2}{q \times \Delta d} \\\\E = \frac{9.11 \times 10^{-31} \times 420^2}{1.602 \times 10^{-19} \times (2-1)} \\\\E = 1. 00 \times 10^{-6} \ N/C$

The charge density is calculated as follows;

$E = \frac{\sigma }{2 \epsilon} \\\\\sigma = E \times 2\epsilon \\\\\sigma = 1.00 \times 10^{-6} \times 2 \times (8.85 \times 10^{-12} )\\\\\sigma = 1.77 \times 10^{-17} \ C/m^2$

Thus, the surface charge density on the sheet is 1.77 x 10⁻¹⁷ C/m².

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