A point charge q = -0.53 nC is fixed at the origin. Where must a proton be placed in order for the electric force acting on it to be exactly opposite to its weight? (Let the y axis be vertical and the x axis be horizontal.)
Express your answer using two significant figures.
To find the position where the electric force acting on the proton is exactly opposite to its weight, we need to consider the forces acting on the proton and set them equal to each other.
The weight of the proton is given by the equation:
Weight = mass * acceleration due to gravity
Since we are given the charge of the proton (q = 1.60 x 10^-19 C), we can find the mass using the relationship between charge and mass:
q = charge of proton
e = elementary charge = 1.60 x 10^-19 C
m = mass of proton
q = (m/e) * e
m = q/e
Next, we need to calculate the electrical force experienced by the proton. The electric force between two charges is given by Coulomb's law:
F = (k * |q₁*q₂|) / r²
In this case, the charge of the fixed point charge is q₁ = -0.53 nC and the charge of the proton is q₂ = 1.60 x 10^-19 C.
Setting the electric force equal to the weight of the proton, we have:
(m * g) = (k * |q₁*q₂|) / r²
where g is the acceleration due to gravity and r is the distance between the point charge and the proton.
Solving for r, we get:
r = sqrt((k * |q₁*q₂|) / (m * g))
Now, we can substitute the given values into the equation and calculate r:
q₁ = -0.53 nC = -0.53 x 10^-9 C
q₂ = 1.60 x 10^-19 C
m = (1.60 x 10^-19 C) / (1.60 x 10^-19 C/e) = 1 e
g = 9.81 m/s²
k = 8.99 x 10^9 Nm²/C² (Coulomb's constant)
Plugging these values into the equation:
r = sqrt((8.99 x 10^9 Nm²/C² * |-0.53 x 10^-9 C * 1.60 x 10^-19 C|) / (1 kg * 9.81 m/s²))
Calculating this, we find that the distance r is approximately 0.0077 meters.
Therefore, the proton must be placed at a distance of approximately 0.0077 meters from the origin (along the x-axis) in order for the electric force acting on it to be exactly opposite to its weight.