Consider the potential field V(x, y) which is 0 and -Vo(Vo>0

[pallab]

Homework Statement

Consider the potential field V(x, y) which is 0 and -Vo (V₀ > 0) respectively in the regions
of y greater and less than zero . Let θ and θ' be the angles of incidence and refraction of
the particle with the y-axis at the point of incidence as it crosses the x-axis . The ratio
sin(θ) / sin(θ ') is given (in terms of Δ = Vo / E) by
region1

region2
(a)√(1+2Vο/E)
(b)√(1+Vο/E)
(c)1+Vο/E
(d)1+2Vο/E

Homework Equations

nsinθ=n'sinθ'
sinθ/sinθ'=v/v'[where v=velocity of particle in 1 & v'in 2]

The Attempt at a Solution

1/2mv²=E=1/2mv^'2-Vο
1/2mv^'2-1/2mv²=Vο
(v^'2 -v²)/v²=2Vο/mv²
(v^'2 /v² )-1=Vο/(1/2mv²)
∴sinθ'/sinθ=√(1+Vο/E)

Which is similar to option b but in question it is sinθ/sinθ'

 

[TSny]

sinθ/sinθ'=v/v' [where v=velocity of particle in 1 & v'in 2]

Are you sure this equation is correct? If Vo is positive, which would be larger: v or v'? θ or θ'?

 

[pallab]

I have just used snell's law.
θ∝v
but I am not sure about increase or decrease of kinetic energy in the presence of potential +ve or -ve.

 

[pallab]

Are you sure this equation is correct? If Vo is positive, which would be larger: v or v'? θ or θ'?

I have just used snell's law.
θ∝v
but I am not sure about increase or decrease of kinetic energy in the presence of potential +ve or -ve.

 

[TSny]

This problem deals with a particle moving from a region of zero potential energy to another region of constant potential energy -Vo.

For this situation, Snell's law as written for light $\frac{\sin \theta_1}{v_1} = \frac{\sin \theta_2}{v_2}$ does not apply. Have you covered how Snell's law is modified for the particle situation?

 

[pallab]

This problem deals with a particle moving from a region of zero potential energy to another region of constant potential energy -Vo.

For this situation, Snell's law as written for light $\frac{\sin \theta_1}{v_1} = \frac{\sin \theta_2}{v_2}$ does not apply. Have you covered how Snell's law is modified for the particle situation?

No.

This problem deals with a particle moving from a region of zero potential energy to another region of constant potential energy -Vo.

For this situation, Snell's law as written for light $\frac{\sin \theta_1}{v_1} = \frac{\sin \theta_2}{v_2}$ does not apply. Have you covered how Snell's law is modified for the particle situation?

No.

 

[TSny]

Then you will need to derive "Snell's law" for the particle.

As a start, consider the x and y components of the velocity of the particle.
Does v_x change when the particle passes through the origin?
Does v_y change?