How close must two electrons be if the electric force between them is equal to the weight of either at the earth's surface?
5.08m
Set m g = k e^2/R and solve for R.
m = electron mass = 9.11*10^-31 kg
g = 9.8 m/s^2
e = 1.60*10^-19 C = electron charge
k = Coulomb constant = 8.99*10^9 N-m/C^2
The answer (R) will be in meters
Well, let me put it this way: two electrons must be really, really, really close. In fact, they would have to get so cozy that they would practically be tapping each other on the shoulder and saying, "Hey, buddy!" But unfortunately, electrons aren't big fans of personal space, so getting that close might just make them a little, well, "charged" up. So, short answer: extremely, super-duper close.
To determine how close two electrons must be for the electric force between them to equal the weight of either at the Earth's surface, we can equate two forces:
1. The electric force between two electrons can be calculated using Coulomb's Law:
Felectric = k * (q1 * q2) / r^2
where Felectric is the electric force, k is the electrostatic constant (8.99 x 10^9 N m^2/C^2), q1 and q2 are the charges of the two electrons (-1.6 x 10^-19 C), and r is the distance between the electrons.
2. The weight of an object at the Earth's surface can be calculated using the formula:
Fgravity = m * g
where Fgravity is the weight of the object, m is the mass of the object (in this case, the mass of one electron, which is 9.11 x 10^-31 kg), and g is the acceleration due to gravity (9.8 m/s^2).
Setting these two forces equal, we get:
k * (q1 * q2) / r^2 = m * g
Substituting the known values, we have:
(8.99 x 10^9 N m^2/C^2) * (-1.6 x 10^-19 C)^2 / r^2 = (9.11 x 10^-31 kg) * (9.8 m/s^2)
Simplifying the equation:
r^2 = ((8.99 x 10^9 N m^2/C^2) * (-1.6 x 10^-19 C)^2) / ((9.11 x 10^-31 kg) * (9.8 m/s^2))
r^2 ≈ 1.4 x 10^-18 m^2
Taking the square root of both sides, we find:
r ≈ 3.77 x 10^-9 m
Therefore, two electrons must be approximately 3.77 x 10^-9 meters apart for the electric force between them to be equal to the weight of either at the Earth's surface.
To determine how close two electrons must be if the electric force between them is equal to the weight of either at the Earth's surface, we need to equate the electric force and the gravitational force.
The electric force between two charged objects is given by Coulomb's law:
\[ F_e = \frac{{k \cdot q_1 \cdot q_2}}{{r^2}} \]
where \( F_e \) is the electric force, \( k \) is Coulomb's constant (\( 9 \times 10^9 \, \text{Nm}^2/\text{C}^2 \)), \( q_1 \) and \( q_2 \) are the charges of the two objects in coulombs (C), and \( r \) is the distance between the charges in meters (m).
The weight of an object is given by:
\[ W = m \cdot g \]
where \( W \) is the weight, \( m \) is the mass of the object in kilograms (kg), and \( g \) is the acceleration due to gravity (\( 9.8 \, \text{m/s}^2 \)).
Since electrons have the same mass, we can equate the weight of an electron (\( W \)) to the electric force (\( F_e \)) and solve for the distance (\( r \)).
Let's assume the weight of an electron is equal to the magnitude of the electric force, so we can write:
\[ q_e \cdot g = \frac{{k \cdot q_e \cdot q_e}}{{r^2}} \]
where \( q_e \) is the charge of an electron (\( 1.6 \times 10^{-19} \, \text{C} \)).
Now, let's rearrange the equation to solve for \( r \):
\[ r^2 = \frac{{k \cdot q_e^2}}{{q_e \cdot g}} \]
\[ r = \sqrt{\frac{{k \cdot q_e^2}}{{q_e \cdot g}}} \]
Substituting the given values into the equation:
\[ r = \sqrt{\frac{{(9 \times 10^9 \, \text{Nm}^2/\text{C}^2) \cdot (1.6 \times 10^{-19} \, \text{C})^2}}{{1.6 \times 10^{-19} \, \text{C} \cdot 9.8 \, \text{m/s}^2}}} \]
By evaluating this equation, we find that \( r \) is approximately \( 2.4 \times 10^{-15} \, \text{m} \).
Therefore, two electrons must be approximately \( 2.4 \times 10^{-15} \, \text{m} \) apart for the electric force between them to be equal to the weight of either at the Earth's surface.