# Question #1e3b6

##### 2 Answers

Option: **Deflection is proportional to charge-mass ratio at a particular instant.**

#### Explanation:

Following is in support of the option selected.

(a) Let

So by conservation energy the particles will have KE equal to the work done on them by the potential gradient.

For proton we can write

Similarly for

We see that velocity of proton is greater than the velocity of alpha particle. If you note closely it is due to different charge to mass ratio of the particles.

**As such proton will always travel faster under the given conditions**.

(b) After gaining above noted velocities under the influence of same PD the particles enter in an uniform electric field (intensity E) and suffer deflections in the direction of E field as both the particles are positively charged.

*Particles are projected along positive direction of x-axis and electric field E is applied towards positive direction of y-axis.*

The force experienced by the proton is

The acceleration of proton

The force experienced by the

Accelertion on

So

Here again we see that proton is accelerated faster as compared to alpha particle in the same electric field E. This leads to conclusion about difference in deflection of the two particles and its dependence on different charge to mass ratios of the particles.

This supports the choice of option already made.

-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.

**Bonus**

Now lets us examine the trajectory to understand the deviations they undergo during their motions in an uniform electric field of intensity E.

**The Trajectory Of Proton**

Let us consider the point of projection as origin and

So it suffers horizontal displacement

Again vertical displacement during this time will be

Dividing (2) by (1) we get

**The Trajectory Of ** **-particle**

If

So it suffers horizontal displacement

Again vertical displacement during this time will be

Dividing (4) by (3) we get

**So the two particles under consideration follow same equation of trajectory**.

**Quantitative explanation**

By eq(2) the vertical displacement of proton in

By eq(4) the vertical displacement of

Dividing (5) by (6)

So in general eq (7) reveals that **the deflection of that particle is more, which has higher charge - mass ratio.**

(In our case this ratio is twice for proton than for

Depends on charge and mass of particle.

#### Explanation:

As both proton and alpha particle are positively charged, both will be deflected in the direction of the electric field. Let this be along the

We know that when a charged particle moves in an electric field the force is given by the expression

We need to remember that as the electric field is orthogonal, there is no change in the velocities of the particles in

Using scalar parts as direction of movement is fixed.

Using Newton's second law of motion

Acceleration

Also we know that Deflection implies change in acceleration of an object due to contact (collision) with a surface or under the influence of a field.

-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.

The acceleration or change in acceleration translates to displacement. Therefore, deduction above can further be quantified by calculating displacement in the

Charge on the proton

Charge on alpha particle

Mass of the proton

Mass of the alpha particle

Let

Let both particles be at rest initially. Suppose origin coincides with the point when the particles enter the uniform electric field. Let displacement for both be zero in the

Now using kinematic equation

For proton

and for alpha particle

Dividing (1) by (2) we obtain

-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.-.

Charge on the proton

Charge on alpha particle

For sake of simplicity lets us take

Therefore equation (3) becomes

We see that for the same time of travel, displacement of proton is twice the displacement of alpha particle which is approximately equal to

Also from above we notice that displacement for proton is more. However, even though applicable for proton-alpha particle example, this is not the correct answer option.