If the distance between two plates of a parallel plate capacitor is doubled, its capacitance

If the area is doubled with the other parameters being constant, then the capacitance would double. If the distance of separation were doubled with the others constant, then the capacitance would be halved.

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To get technical, you want to look at Coulomb's law. This states that

"The magnitude of the Electrostatics force of interaction between two point charges is directly proportional to the scalar multiplication of the magnitudes of charges and inversely proportional to the square of the distances between them." - Wikipedia

The formula for this is:

\$F = k_e \frac{q_1 q_2}{r^2}\$

Where \$F\$ is the electrostatic force between two charges, \$k_e\$ is a 'proportionality constant' (eg the dielelectric constant in a capacitor), and \$r\$ is the distance between the two charges \$q_1\$ and \$q_2\$.

There are other forms of the equation - such as this one specifically for an electric field:

\$E = \frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\$

Which tells us the force at a distance \$r\$ from the single point charge \$q\$.

If you want to start getting really technical then you need to start reading up on quantum mechanics and the interactions between particles and the energies involved in it.

When two particles (say electrons in this case) interact they send quantum particles between them (photons). These, like the rats in the basement, require energy to move. The greater the distance the higher the energy. The higher the energy taken to move the photons the lower the charge left between the two plates.

That's a very simplistic view of it and there is one helluva lot more detail in there to be discovered - such things as Quantum Tunneling, Leptons, Fermions, Bosons, etc. It's fascinating reading if you have the time. I'd recommend Steven Hawking's A Brief History of Time as a good starting point. Follow that up with F. David Peat's Superstrings and the Search for the Theory of Everything and you won't go far wrong. While both these books are getting a bit long in the tooth now and the theories are all still evolving, they give good insights into the workings of the universe at a subatomic level.

If the distance between the plates of the parallel plate capacitor is halved and the dielectric constant of dielectric is doubled, then its capacity will increase by 4 times.

Explanation:

C = `("k"epsilon_0"A")/"d" ∝ "k"/"d"`

Hence,  `"C"_1/"C"_2 = "k"_1/"k"_2 xx "d"_2/"d"_1 = "k"/(2"k") xx ("d""/"2)/"d" =  1/4`

Therefore, C2 = 4C1

OR

`"I"/2  "C"  ("V"_2^2 - "V"_1^2)`

Hence, `1/2 xx 15 xx 10^-6 xx (25^2 - 15^2)`

Therefore, `"C"_2 = 4"C"_1`