System of conducting spheres

In summary: So the ratio for sphere a would be a2/(a2+b2+c2). The charge on sphere a would then be Q multiplied by that ratio. Similarly, the charge on sphere b would be Q multiplied by the ratio b2/(a2+b2+c2). The charge on sphere c would be the remaining charge, Q multiplied by the ratio c2/(a2+b2+c2). Therefore, the total charge Q for the system of three spheres would be the sum of these three charges. In summary, the total charge Q for the system of three conducting spheres connected by thin wires can be determined by calculating the surface area for each sphere and using it to determine the ratio of charge on each sphere
  • #1
mitleid
56
1
Was curious how some of you guys would solve this problem...

Three conducting spheres of radii a, b and c are connected by negligibly thin conducting wires. Distances between the spheres are much larger than their sizes. The electric field on the surface of a is measured to be E[tex]_{a}[/tex]. What is the total charge Q that this system of three spheres holds?

E = Q/r[tex]^{2}[/tex]*Ke

Q = Q[tex]_{a}[/tex]+Q[tex]_{b}[/tex]+Q[tex]_{c}[/tex]

The way I solved it is most likely not the way my professor intended. I said that since the amount of charge on each sphere is a function only of the radius of the sphere...

a + b + c = x

a/x = percentage of Q shared on sphere a (called this S[tex]_{a}[/tex])

so Q[tex]_{a}[/tex] = S[tex]_{a}[/tex]*Q
and Q = Q[tex]_{a}[/tex]/S[tex]_{a}[/tex]

I imagine I'm missing a conceptual link that'd make another path to solving this more clear.
 
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  • #2
The charge surface density would equal, so the total Q is distributed according to the fraction of surface area.

Determine the surface area for each sphere and total of all three, then ratio the area of each sphere to the total.

Area of sphere is proprotional to r2, where r is the radius.
 
  • #3


I would approach this problem using the principles of electrostatics and Coulomb's law. Since the distances between the spheres are much larger than their sizes, we can assume that the electric field on the surface of each sphere is uniform and radial, and that the spheres are all at the same potential.

Using Coulomb's law, we can calculate the electric field at any point on the surface of a sphere as:

E = Q/r^2 * Ke

Where Q is the charge on the sphere, r is the radius of the sphere, and Ke is the Coulomb's constant.

We can also use Gauss's law to relate the electric field and the charge enclosed within a closed surface. Since the electric field is uniform on the surface of each sphere, we can use a spherical Gaussian surface to enclose each sphere individually.

For sphere a, the electric field is given as E_a = Q_a/a^2 * Ke. Similarly, for spheres b and c, we have E_b = Q_b/b^2 * Ke and E_c = Q_c/c^2 * Ke.

Since the spheres are connected by conducting wires, they are all at the same potential. This means that the potential difference between any two spheres is the same. We can use this fact to write the following equations:

E_a = E_b = E_c

Q_a/a^2 * Ke = Q_b/b^2 * Ke = Q_c/c^2 * Ke

Simplifying, we get:

Q_a/a^2 = Q_b/b^2 = Q_c/c^2

Solving for Q_a, Q_b, and Q_c, we get:

Q_a = Q_b * (a/b)^2

Q_a = Q_c * (a/c)^2

Substituting these expressions into the equation Q = Q_a + Q_b + Q_c, we get:

Q = Q_b * (a/b)^2 + Q_c * (a/c)^2 + Q_b

Since the total charge Q is a function of the radii of the spheres, we can write it as:

Q = f(a, b, c)

To solve for Q, we need to know the values of a, b, and c. However, we are given the electric field on the surface of sphere a, E_a. We can use this information to find the ratio of the charges on the spheres:

E_a = Q_a/a^
 

1. What is a system of conducting spheres?

A system of conducting spheres is a collection of multiple conductive spheres that are connected together. This allows for the transfer of electrical charge between the spheres, creating an electrically charged system.

2. How is the charge distributed in a system of conducting spheres?

The charge in a system of conducting spheres is distributed evenly among all the spheres. This is known as the principle of charge distribution, where the charge is evenly spread out on the surface of each sphere.

3. What is the significance of a system of conducting spheres?

A system of conducting spheres is a simplified model used in electrostatics to study the behavior of charged particles. It helps in understanding the concepts of electric charge, electric potential, and electric fields.

4. Can the distance between spheres affect the behavior of a system of conducting spheres?

Yes, the distance between spheres can affect the behavior of a system of conducting spheres. As the distance between spheres decreases, the electric potential and electric field increases, leading to a stronger interaction between the spheres.

5. How does the charge on one sphere affect the charge on the other spheres in a system of conducting spheres?

In a system of conducting spheres, the charge on one sphere can affect the charge on the other spheres through the transfer of electrical charge. This can lead to a redistribution of charge among the spheres, depending on the arrangement and distance between them.

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