Charge on the dust particle: [tex]-6.25\cdot 10^{-13} C[/tex]
Explanation:
In order for the dust particle to be in equilibrium, the electric force acting on it must balance the weight of the particle, so:
[tex]qE=mg[/tex] (1)
where
q is the charge on the dust particle
E is the magnitude of the electric field at the location of the particle
[tex]m=4.5\mu g = 4.5\cdot 10^{-9} kg[/tex] is the mass of the particle
[tex]g=9.8 m/s^2[/tex] is the acceleration of gravity
To find q, we need to calculate the electric field generated by the flat carpet first. Assuming it to be a flat infinite sheet, then the electric field it produces is
[tex]E=\frac{\sigma}{2\epsilon_0}[/tex]
where
[tex]\sigma=\frac{Q}{A}[/tex] is the charge density on the carpet, where
[tex]Q=10 \mu C = 10\cdot 10^{-6} C[/tex] is the charge on the carpet
[tex]A=(2.0m)(4.0 m)=8 m^2[/tex] is the area of the carpet
[tex]\epsilon_0 = 8.85\cdot 10^{-12} F/m[/tex] is the vacuum permittivity
Substituting,
[tex]E=\frac{Q}{2A\epsilon_0}=\frac{10\cdot 10^{-6}}{2(8.0)(8.85\cdot 10^{-12})}=7.06\cdot 10^4 N/C[/tex]
Now we can finally re-arrange eq.(1) to find the charge on the dust particle:
[tex]q=\frac{mg}{E}=\frac{(4.5\cdot 10^{-9})(9.8)}{7.06\cdot 10^4}=6.25\cdot 10^{-13} C[/tex]
Note that the charge must be negative, because the charge on the carpet is negative, so the electric field points downward, so for the dust particle in order to be repelled its charge must be negative also.
Learn more about electric fields:
brainly.com/question/8960054
brainly.com/question/4273177
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