After rubbing a balloon against someone's hair, a balloon has accumulated a static electric charge. The air inside the balloon has the same density as the air outside of the balloon, and the balloon itself
weighs 1.41mN. How many free electrons are there on the surface of the balloon if an electric field of magnitude 4.97x10^11N/C is required to keep the balloon floating at a constant height?
To find the number of free electrons on the surface of the balloon, we can use the electric force equation:
F = q * E
where F is the electric force, q is the charge, and E is the electric field.
First, let's find the charge on the surface of the balloon. Since the air inside and outside the balloon have the same density, the buoyant force acting on the balloon is equal to its weight.
F_buoyant = F_weight
The weight of the balloon is given as 1.41 mN (millinewtons), so we can convert it to Newtons:
F_weight = 1.41 x 10^(-3) N
Since electric force and weight are both forces, they must be equal. Therefore, we have:
F_weight = F = q * E
Now, let's substitute the value for the electric field into the equation:
1.41 x 10^(-3) N = q * 4.97 x 10^11 N/C
To solve for q, we divide both sides of the equation by the electric field:
q = (1.41 x 10^(-3) N) / (4.97 x 10^11 N/C)
q ≈ 2.83 x 10^(-15) C
Now, we know that the charge on an electron is approximately -1.6 x 10^(-19) C. To find the number of electrons, we divide the total charge (q) by the charge of a single electron:
Number of electrons = q / charge of an electron
Number of electrons = (2.83 x 10^(-15) C) / (-1.6 x 10^(-19) C)
Number of electrons ≈ -1.77 x 10^4
Since the number of electrons cannot be negative, we take the absolute value and get approximately 1.77 x 10^4 free electrons on the surface of the balloon.
Therefore, there are approximately 1.77 x 10^4 free electrons on the surface of the balloon.