Two conducting spheres are isolated from their surroundings. The radii of spheres A and B are R and R/2 , respectively. Initially sphere A has charge +Q , and sphere B is neutral. A wire is connected between the sphere. What is the final charge on each sphere after equilibrium is reached and why?
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To find the final charge on each sphere after equilibrium is reached, we can apply the principle of charge conservation and use the concept of charge redistribution between two connected conductors.
1. Initially, sphere A has charge +Q and sphere B is neutral.
2. When the wire is connected between the spheres, electrons can flow from sphere A to sphere B through the wire until they both reach equilibrium.
3. Since both spheres are conductors, the charges distribute themselves on the outer surface of the spheres and the net charge inside the spheres is zero.
4. The amount of charge transferred from sphere A to sphere B is determined by the ratio of their radii.
- The ratio of the final charges Qa and Qb on the two spheres is given by Qa/Qb = -Ra/Rb, where Ra and Rb are the radii of spheres A and B, respectively.
- In this case, Ra = R and Rb = R/2, so Qa/Qb = -R/(R/2) = -2.
5. Since the initial charge on sphere A is +Q, the final charge on sphere B (Qb) can be calculated as Qb = -2Q.
6. Similarly, the final charge on sphere A (Qa) can be calculated as Qa = -2Qa (using the charge conservation principle).
7. Therefore, the final charge on sphere A is -2Q, and the final charge on sphere B is -Q.
In conclusion, the final charge on sphere A is -2Q and the final charge on sphere B is -Q after equilibrium is reached because charges redistribute themselves such that the ratio of the final charges is determined by the ratio of their radii.
To determine the final charge on each sphere after equilibrium is reached, we need to analyze the charge transfer between the two spheres.
Initially, sphere A has a charge of +Q, and sphere B is neutral. When the wire is connected between the spheres, the charge will redistribute in order to achieve equilibrium.
Let's consider the process step by step:
1. Due to the difference in radii, the electric field at the surface of sphere A is stronger than at the surface of sphere B. This means that the electric potential is higher on sphere A.
2. As a result of the potential difference, electrons will flow from sphere B to sphere A through the wire until the potentials of both spheres are equal. The positive charge on sphere A attracts the negative charges (electrons) from sphere B.
3. The charge transfer will continue until the potentials of both spheres are equal, indicating that equilibrium has been reached. At this point, no further charge transfer will occur.
To determine the final charge on each sphere, we need to apply the principle of charge conservation. The total charge before and after the transfer must be the same.
Initially, sphere A has a charge +Q, and sphere B is neutral. During the charge transfer, electrons are transferred from sphere B to sphere A.
The final charge on sphere A will be +Q + (-q), where (q) is the charge transferred from sphere B. The negative sign indicates that electrons have been added to sphere A.
The final charge on sphere B will be 0 - (-q), which simplifies to +q. The positive charge indicates that sphere B has lost electrons.
Therefore, after equilibrium is reached, sphere A will have a final charge of +Q - q, and sphere B will have a final charge of +q.
It's important to note that the magnitude of the charge transferred will depend on various factors such as the radii of the spheres and the initial charge on sphere A. To determine the exact values, you would need to know the specific values of these variables and apply appropriate formulae.