# Homework Help: Charged Ballon

1. Feb 13, 2012

### vivekrai

Problem : A conducting metal balloon of Radius ' R ', essentially of negligible mass is charged with charge ' Q '. It has to carry a payload of mass ' M '. Find how the charge ' Q ' depends on the mass M and the Intrinsic Energy ' W ' of the balloon.

Attempt:

We have $P_{atm} = \frac{σ^{2}}{2ε_0}$ for the balloon to sustain the charge on it. W is also known. Now by Archimedes Principle, the Force of buoyancy is equal to the weight of the air displaced by the balloon which is $F_b = \frac{4}{3} π R^3 \ge Mg$.

But How does the charge and the Energy W influence the motion. Kindly suggest me. I do not need computation help.

2. Feb 15, 2012

### vivekrai

3. Feb 16, 2012

### Staff: Mentor

I don't understand how charge would enter into the problem at all. It would have to rely on electrostatic repulsion somehow, but only the balloon is charged in your question.

4. Feb 16, 2012

### Staff: Mentor

The static charge is self-repellent (which is why it resides on the surface of a conducting body), so it will create an effective outward pressure. What I don't quite see is the "intrinsic energy 'W' of the balloon". Perhaps this is related to tension and elasticity of the balloon.

If the balloon was initially filled and sealed containing 1 ATM of pressure, the static charge would enlarge the volume...

5. Feb 17, 2012

### vivekrai

Intrinsic Energy is basically the energy required to charge the sphere , which is in this case $W=\frac{Q^2}{8\pi\epsilon_0 R}$.

Since, the balloon is initially completely evacuated, it is charged with Q so as to create an electrostatic pressure which balances the atmospheric Pressure.
The Balloon is supposed to carry the Payload based on Archimedes Principle.

I have cleared this thing above.

Now my Doubt is whether Charge residing on it surface or the ' Intrinsic ' Energy possessed by it is going to affect its motion based on Archimedes Principle?

6. Feb 17, 2012

### Staff: Mentor

I think you'll have to first determine the required balloon radius R to support the weight of cargo mass M. With the balloon radius compute the surface area. Next determine the required charge density to support 1atm pressure. Use the surface area and charge density to determine the charge required.

I don't see how the 'intrinsic energy' will play a role in this.

7. Feb 17, 2012

### vivekrai

Okay.. so we have got a Relation with the charge density on the surface and that too because we have to balance the requisite atmospheric Pressure. It is clear.

Now regarding ' Intrinsic Energy ' - It has dependence because W depends on R which in turn affects the charge density on the surface. So these are clear to me..

We suppose that air has some conductivity σ as a result of which the charge on it leaks away. To stop this we install some mechanism which maintains a constant charge Q0 on the balloon. Let us denote the power of it with .

Now, How does the Payload M depends on ∏ ?

8. Feb 17, 2012

### Staff: Mentor

The leakage current will depend upon the surface area, potential of the surface, the conductivity of the air, and the ambient potential of the surrounding air. Once the balloon is of the required size to handle the load, the parameters are fixed.

9. Feb 17, 2012

### vivekrai

Could you help me a bit with finding the Power for it? I have found the expression for leakage charge as $Q = Q_o e^{-\frac{\sigma t}{\epsilon_0 }}$ , where Q is charge remaining on balloon after time t and Q0 the initial charge.

10. Feb 17, 2012

### Staff: Mentor

It seems to me that you want to maintain a constant charge, and thus constant voltage on the balloon. If the initial charge is Q0, then determine dQ/dt at time t = 0. That'll be the constant current that you need to supply to maintain the charge. Given voltage and current you can find the power.

11. Feb 17, 2012

### vivekrai

We need either of them or both (Current/Voltage) ? Because power = I^2R pr V^2/R ? Could you please provide me with an expression for this one?

12. Feb 17, 2012

### Staff: Mentor

If you can determine the net resistance of the leakage path then you can use the balloon voltage and that resistance to find the power. Otherwise, if you have the current and the voltage, then P = V*I.

I assumed that you'd go for the current if you have the required parameters for the charge vs time equation that you quoted.

13. Feb 17, 2012

### vivekrai

In that case, the Initial Current and which has to be maintained is given by $I = Q_0 \frac{\sigma}{\epsilon_0}$ and the Constant Potential of the balloon is $V=\frac{Q_0}{4\pi\epsilon_0 R}$.

Hence $\Pi = \frac{Q^2_0\sigma}{4\pi\epsilon^2_0 R}$ ? Is it correct?

Last edited: Feb 17, 2012
14. Feb 18, 2012

### Staff: Mentor

It looks okay thus far. Are you going to resolve R and Q0 according to the cargo load requirement, etc., and then substitute their expressions for R and Q0 into this power equation?

15. Feb 19, 2012

### vivekrai

R is fixed. I have to resolve it for the charge Q . The cargo is also Fixed.

16. Feb 19, 2012