Can someone please explain energy of electrons in Bohr's model?

Question

Energy of and electron is $$E = \text{kinetic energy} + \text{negative of potential energy}. \tag{eq-1}$$ But energy of electron in the $n$th orbital is also $$E = -\frac{13.6}{ n^2} \tag{eq-2}$$ Which means that the energy of the electron would get less and less negative as we go to further orbitals, which means that the kinetic energy of the electron is increasing (from eq-1). However that is counter-intuitive because as the electron goes from lower orbital to higher its going from a lower potential (closer to the nucleus) to a higher potential (away from nucleus i,e. against the electric field). And thus the potential energy should increase and the total energy should be more negative.

Another thing which is mind boggling is that the velocity of electron in the $n$th orbit is, $$v = \frac{e^2}{nh\epsilon_0}$$ which suggests that velocity decreases as we go from inner to outer orbitals, which means that kinetic energy decreases! All this does not make sense to me. I know I made a mistake somewhere so if someone can help me please explain.

At the same time my textbook says that the outer energy level electrons would have more energy so, they would need less energy to be excited. But what energy do they have? The velocity equation suggests that outer electrons have less and less kinetic energy as we go to oter orbitals and eq 2 suggest that the electron even decrease in potential energy as they go in further energy levels(which should increase)

Answer 1

You write

E = kinetic energy + negative of potential energy. (eq-1)

and then

as the electron goes from lower orbital to higher […] the potential energy should increase and the total energy should be more negative.

Remember that the electrostatic potential energy between two charges $q_1,q_2$ separated by some distance $r$ is

$$ U = \frac{1}{4\pi\epsilon_0}\frac{q_1 q_2}{r} $$

This potential is not strictly positive: it has the same sign as the product of charges $q_1\cdot q_2$. From the definition of the force $\vec F = -\vec\nabla U$ (or, in one dimension, $\vec F = -\hat x\frac{\mathrm d}{\mathrm dx}U$), you should be able to convince yourself that this sign convention is required to make opposite charges attract and like charges repel.

When an electron moves from a lower to a higher orbit, the potential energy increases by becoming less negative. Your equation (eq-1) is wrong. The total energy is kinetic plus potential; it’s just that the potential energy between opposite charges happens to be negative.