# How many electron charges are there on the drop? (Millikan)

## Homework Statement

In a measurement of the electron charge by Millikan's method, a potential difference of 1.5 kV can be applied between horizontal parallel metal plates 12 mm apart. With the field switched off, a drop of oil of mass 10-14 kg is observed to fall with constant velocity 400 μm s-1. When the field is switched on, the drop rises with constant velocity 80 μm s-1. How many electron charges are there on the drop? (You may assume that the air resistance is proportional to the velocity of the drop, and that air buoyancy may be neglected.)

(The electronic charge = 1.6 * 10-19 C, the acceleration due to gravity = 10 m s-2.)

2. The attempt at a solution
As I understand I need to find Q?

In my book I have this formula: Q = 6 π r η v / E, where η is the coefficient of viscosity of air. The formula is applicable "when an electric field has been applied such that the drop is stationary." There are also formulas when there is no electric field, but I think in this case we have electric field.

But we don't have the radius and have the distance between plates instead.

I am also not sure whether I need to find Q, or some other value. Also how to find η?

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gneill
Mentor
As I understand I need to find Q?

In my book I have this formula: Q = 6 π r η v / E, where η is the coefficient of viscosity of air. The formula is applicable "when an electric field has been applied such that the drop is stationary." There are also formulas when there is no electric field, but I think in this case we have electric field.

But we don't have the radius and have the distance between plates instead.

I am also not sure whether I need to find Q, or some other value. Also how to find η?
Yes you need to find Q. In particular you need to find out how many elementary charges it takes to total up to Q.

You are not given enough information to calculate the drag force directly. However, you are told to assume that it is proportional to velocity. Can you determine one value for the drag force at a particular velocity (other than zero velocity, of course)? Can you then scale it for other velocities?

moenste
Yes you need to find Q. In particular you need to find out how many elementary charges it takes to total up to Q.

You are not given enough information to calculate the drag force directly. However, you are told to assume that it is proportional to velocity. Can you determine one value for the drag force at a particular velocity (other than zero velocity, of course)? Can you then scale it for other velocities?
Q = (6 π η / E) * (9 η v / 2 ρo g)1 / 2 v

E = V / d = 1500 / 0.012 = 125 000 V m-1.

Now η (coefficient of viscosity of air) and ρo (density of oil) are required.

Update
As I understand when there is no electric field velocity is 400 μm s-1. Weight = Upthrust due to air + Viscous drag. Upthrust is zero. So: (4/3) π r3 po g = 6 π r η v. Simplify them to find radius: r = √ 4.5 η v / ρo g.

Then we find radius and turn the field on. We already found E = 125 000 V m-1. And then we use Q = (4/3) π r3 ρo g / E. Q = 4 π (√ 4.5 η v / ρo g)3 ρo g / 3 E.

Only need to find η and ρo somehow...

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gneill
Mentor
You can forget the formula for drag and all of its constants that you don't know. Just find the force (in Newtons) that it creates when the oil drop is falling at a constant speed without the electric field in place.

moenste
You can forget the formula for drag and all of its constants that you don't know. Just find the force (in Newtons) that it creates when the oil drop is falling at a constant speed without the electric field in place.
Hm, is it F = m g? We have the mass present.

gneill
Mentor
Hm, is it F = m g? We have the mass present.
Yes. The drop is falling with constant velocity so the net force acting is zero.

moenste
Yes. The drop is falling with constant velocity so the net force acting is zero.
F = m g
E = V / d

m g = Q (V / d)
Q = m g d / V = 10-14 * 10 * 0.012 / 1500 = 8 * 10-19 C.

gneill
Mentor
F = m g
E = V / d
You want to specify that that force F above is the force due to drag for that one velocity only.
m g = Q (V / d)
Q = m g d / V = 10-14 * 10 * 0.012 / 1500 = 8 * 10-19 C.
Can you explain in words what the first equation above represents? I can see an electric force, I can see the gravitational force, but I don't see the drag accounted for.

moenste
You want to specify that that force F above is the force due to drag for that one velocity only.

Can you explain in words what the first equation above represents? I can see an electric force, I can see the gravitational force, but I don't see the drag accounted for.
Weigth = Upthrust + Electric force
Weight, as you say, shouldn't be 4/3 π r3 ρo g but F = m g.

Upthrust is zero.

Electric force = Q E, Q = charge on the drop, E = the electric field strength.

gneill
Mentor
There will be a different drag force acting when the drop is moving upwards because its moving at a different speed than when it was falling. You need to account for that drag force. Use the given assumption about the drag force and velocity.

moenste
There will be a different drag force acting when the drop is moving upwards because its moving at a different speed than when it was falling. You need to account for that drag force. Use the given assumption about the drag force and velocity.
What is drag force?

gneill
Mentor
What is drag force?
Air resistance, or viscous drag. You calculated a value for the drag force for the falling oil drop in posts #5 and #7.

moenste
Air resistance, or viscous drag. You calculated a value for the drag force for the falling oil drop in posts #5 and #7.
I'm sorry, I don't get it.

I need to find Q. Assuming everything I did is not correct. What do I need to do then?

Update
I think I understand what you mean. If we have no electric field then the drop is falling and therefore weight = m g. And therefore we have weight = upthrust due to air + viscous drag (in contrast to when there is an electric field and weight = upthrust + electric force). I used to formula for the situation when an electric field is present.

So now we have weight = viscous drag. m g = 6 π r η v. Though I don't see how it helps -- we need the radius and the coefficient of viscosity of air η.

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gneill
Mentor
Yes, you need to find Q. You do that by finding out what force that charge experiences due to the known electric field. But there are several forces acting at once and you need to factor out their contributions. Gravity is easy because you have the mass of the oil drop. Air resistance also acts on the drop whenever it is moving through air. It's a type of friction.

The idea is to establish what this friction force is for the upward moving drop so that you can "remove it" from the sum of forces, just as you will "remove" the gravitational force.

That's why you went to the trouble of finding what the air resistance force is for the drop while it was falling at constant speed. The only two forces acting then were gravity and this friction force.

Now you need to find the friction force acting on the upward moving drop so that you can account for it in the net force sum.

moenste
Yes, you need to find Q. You do that by finding out what force that charge experiences due to the known electric field. But there are several forces acting at once and you need to factor out their contributions. Gravity is easy because you have the mass of the oil drop. Air resistance also acts on the drop whenever it is moving through air. It's a type of friction.

The idea is to establish what this friction force is for the upward moving drop so that you can "remove it" from the sum of forces, just as you will "remove" the gravitational force.

That's why you went to the trouble of finding what the air resistance force is for the drop while it was falling at constant speed. The only two forces acting then were gravity and this friction force.

Now you need to find the friction force acting on the upward moving drop so that you can account for it in the net force sum.
Not sure whether you saw this:
Update
I think I understand what you mean. If we have no electric field then the drop is falling and therefore weight = m g. And therefore we have weight = upthrust due to air + viscous drag (in contrast to when there is an electric field and weight = upthrust + electric force). I used to formula for the situation when an electric field is present.

So now we have weight = viscous drag. m g = 6 π r η v. Though I don't see how it helps -- we need the radius and the coefficient of viscosity of air η.
Is this correct logic?

gneill
Mentor
Is this correct logic?
Yes the logic is fine as far as it goes. But you don't need to know any of the constants for the air resistance formula; it's not needed here since you can use the value for the force at a particular velocity to find the force for other velocities thanks to the given assumption about how the air resistance varies with velocity.

moenste
Yes the logic is fine as far as it goes. But you don't need to know any of the constants for the air resistance formula; it's not needed here since you can use the value for the force at a particular velocity to find the force for other velocities thanks to the given assumption about how the air resistance varies with velocity.
OK, so no electric field: Weight = Upthrust (zero) + Air resistance → m g = Air resistance.

With electric field: Weight = Upthrust (zero) + Electric force → 4 / 3 π r3 ρo g = Q E → 4 / 3 π r3 ρo g = Q V / d.

I think this should be correct.

gneill
Mentor
You're still plugging in the useless air resistance formula. You need to use the first air resistance force to find a value for the second. Then write your force balance using that.

moenste
You're still plugging in the useless air resistance formula. You need to use the first air resistance force to find a value for the second. Then write your force balance using that.
In that case I'm missing some force.

This is what I have my book:

From book:
No EF: Weight = Upthrust due to air + Viscous drag.
EF: Weight = Upthrust + Electric force.

So:
No EF: Viscous drag = 10-14 * 10 = 10-13 N.
EF: Q = m g / (V / d) = 10-13 / (1500 / 0.012) = 8 * 10-19 C.

So we have the answer Q = 8 * 10-19 C. I don't understand why we need all the other numbers given in the problem and why we need to calculate the no EF force.

gneill
Mentor
In the book's figure the drop on the right hand side is stationary, so no air resistance force, just the electric force balancing the gravitational force.

In the present problem the oil drop is moving in both cases.

moenste
In the book's figure the drop on the right hand side is stationary, so no air resistance force, just the electric force balancing the gravitational force.

In the present problem the oil drop is moving in both cases.
Aaaah, it even says so.

So, in that case:
No EF: Air resistance = 10-14 * 10 = 10-13 N.
EF: Weight = Upthrust + Electric force + Air resistance → 10-13 N = zero + Q E + 10-13 N → Q E = 0?

gneill
Mentor
I don't see where you've determined the air resistance force for the new oil drop speed. The speed is different than the first case so the air resistance must be different. Use the first case result to find the value for the new speed.

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moenste
I don't see where you've determined the air resistance force for the new oil drop speed. The speed is difference than the first case so the air resistance must be different. Use the first case result to find the value for the new speed.
Maybe something like Air resistanceFalling * SpeedFalling = ARRising * SpeedRising → 10-13 * (400 * 10-6) = ARRising * (80 * 10-6) → ARRising = 5 * 10-13 N?

Weight = Upthrust + EF + ARRising
10-13 N = zero + Q E + 5 * 10-13 N
Q E = -4 * 10-13 N
Q = -4 * 10-13 / 125 000 = -3.2 * 10-18 Q.

gneill
Mentor
Let's concentrate on the air resistance for a moment.

In the first case you have:
v1 = 400 (micrometers per second)
F1 = Mg = 10-13 N

Now you are given that air resistance is proportional to velocity. Mathematically: F ∝ v

Write that as an equality by inserting a proportionality constant and solve for that constant. Then you can determine the F for any given v.

moenste
Let's concentrate on the air resistance for a moment.

In the first case you have:
v1 = 400 (micrometers per second)
F1 = Mg = 10-13 N

Now you are given that air resistance is proportional to velocity. Mathematically: F ∝ v

Write that as an equality by inserting a proportionality constant and solve for that constant. Then you can determine the F for any given v.
v1 = 4 * 10-4 m s-1.
F1 = m g = 10-14 * 10 = 10-13 N.

v2 = 8 * 10-5 m s-1.
F2 = ?

F1 = k v1, where k = F1 / v1 = 10-13 / 4 * 10-4 = 2.5 * 10-10.

So F2 = k v2 = 2.5 * 10-10 * 8 * 10-5 = 2 * 10-14 N.

---

Weight = Upthrust + EF + ARRising or F2
10-13 N = zero + Q E + 2 * 10-14 N
Q E = 8 * 10-14 N
Q = 8 * 10-14 / 125 000 = 6.4 * 10-19 Q.

Then 6.4 * 10-19 / 1.6 * 10-19 = 4, not 6 as in the answer. What's wrong? I recalculated everything and checked the book answer, it's 6. So, don't think that it's the calculation.

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