Relationship Between Charge, Mass, and Balloon Intrinsic Energy

[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.

 

[vivekrai]

Help Please!

 

[berkeman]

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.

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.

 

[gneill]

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.

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...

 

[vivekrai]

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.

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.

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".

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?

 

[gneill]

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?

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.

 

[vivekrai]

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

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..

Addition to the question
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 ∏ ?

 

[gneill]

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..

Addition to the question
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 ∏ ?

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.

 

[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.

 

[gneill]

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.

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.

 

[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?

 

[gneill]

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?

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.

 

[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?

 

[gneill]

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?

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?

 

[vivekrai]

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

 

[vivekrai]

One Addition

Now the balloon is assumed to have tiny holes having a total area S << R². Now Let P1 is the atmospheric Pressure and P2 << P1 , the pressure of the inside of balloon. The air enetrs into the balloon through the holes. We have to find the velocity of gas when it enters the balloon. Temperature of ambient is T1 , adiabatic index γ , molar mass η .

Attempt at Solution

Obviously, Since the gas enters through the holes so rapidly that it doesn't have time to exchange heat with the atmosphere i.e., the process is adiabatic. Also to find the velocity, we need to use the bernoulli equation.

So, E_i = E_f for an air element. E_f = 1/2 η v² . But what do i take for E_i --> the initial energy. Also How do I use the information that I got for adiabatic thing.