(a) What is the electric potential energy in joules of two electrons separated by 1.98 nm? (b) What would be it be if the separation was doubled?
(a) Oh, electrons and their potential energy, they're quite the dynamic duo. Let's calculate it, shall we? The electric potential energy between two charges is given by the equation U = k * (q1 * q2) / r, where k is Coulomb's constant (8.99 x 10^9 N m^2 / C^2), q1 and q2 are the charges (both electrons have a charge of -1.6 x 10^-19 C), and r is the separation distance (1.98 nm = 1.98 x 10^-9 m). Crunching the numbers, we get U = 3.611 x 10^-18 J. Voila!
(b) Ah, now we're doubling the separation. That's like trying to socially distance two electrons even more. Well, if we double the separation distance to 2 * 1.98 nm, we'll simply plug it into our formula and recalculate. So U = k * (q1 * q2) / (2 * r) = (3.611 x 10^-18 J) / 2 = 1.8055 x 10^-18 J. Double the separation, but only half the potential energy. Talk about an energy-efficient electron relationship!
To calculate the electric potential energy between two electrons, we can use Coulomb's law, which states that the electrostatic force between two point charges is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
(a) Let's calculate the electric potential energy for two electrons separated by a distance of 1.98 nm.
First, we need to convert the distance from nanometers to meters:
1 nm = 1 × 10^(-9) m
So, the separation between the electrons is:
1.98 nm × (1 × 10^(-9) m/nm) = 1.98 × 10^(-9) m
The charge of an electron is approximately -1.6 × 10^(-19) Coulombs.
Now, we can use Coulomb's law to calculate the electric potential energy:
Electric potential energy (U) = (k * q1 * q2) / r
where:
- U is the electric potential energy
- k is the electrostatic constant (approximately 9 × 10^9 Nm^2/C^2)
- q1 and q2 are the charges of the two electrons
- r is the separation distance between the electrons
Substituting the values:
U = (9 × 10^9 Nm^2/C^2) * (-1.6 × 10^(-19) C) * (-1.6 × 10^(-19) C) / (1.98 × 10^(-9) m)
Calculating this expression will give you the electric potential energy in joules.
(b) To find the electric potential energy if the separation is doubled, we need to use the new separation distance and recalculate:
New separation distance = 2 * 1.98 × 10^(-9) m
Using the new separation distance in the electric potential energy formula will give you the updated value.