a) An infinite number of changes each equal to q are placed along the x-axis at x = 1, x = 2, x = 4, x = 8, . . . and so on. Find the potential and the esoteric at x = 0 due to this set of changes. What will be the potential and electric field if, in the above set-up, the corrective charges have opposite sign?
(b) Two metal spheres, each of mass 1 gm and radius 0.5 cm are placed on a smooth, horizontal, insulated plate such that their centres are 15 cm apart. One is charged to a potential 500 volt and the other to 1000 volt. What is the velocity of each sphere when they drift away to a distance of 30 cm. between their centres?
a) To find the potential at x = 0 due to an infinite number of charge changes placed along the x-axis, we can use the concept of the electric potential due to point charges. Each charge, q, placed at x = 1, x = 2, x = 4, x = 8, and so on, will contribute to the total electric potential at x = 0.
The potential due to a single charge q at a distance x from the point x = 0 can be calculated using the equation:
V = k * q / x
Where V is the potential, k is the electrostatic constant, q is the charge, and x is the distance from the point charge.
Since there are infinite charges, we need to sum up the contributions from all the charges. However, since the charges are placed at positions corresponding to powers of 2 (x = 1, 2, 4, 8, ...) we can take advantage of a geometric series to calculate the sum.
The sum S of an infinite geometric series with a common ratio r (in this case, r = 2) is given by:
S = a / (1 - r)
In our case, a = q and r = 2, so the sum of the infinite geometric series is:
S = q / (1 - 2) = q / (-1) = -q
Therefore, the potential at x = 0 due to this set of charges is -q.
To find the electric field (esoteric) at x = 0, we can use the relationship between electric field and potential. The electric field is the negative derivative of the potential with respect to x.
In our case, since the potential is constant (-q), the electric field at x = 0 will be zero.
If the corrective charges have the opposite sign, the potential at x = 0 due to this set of charges will also be -q, as the individual potential contributions from each charge will be negative. However, since the charges have opposite signs, the overall potential will still be negative.
The electric field at x = 0 when the corrective charges have opposite signs will also be zero, as the charges cancel each other's contribution to the electric field.
b) To calculate the velocity of each sphere when they drift away to a distance of 30 cm between their centers, we can use the principle of conservation of energy.
Given:
Mass of each sphere = 1 gm
Radius of each sphere = 0.5 cm
Potential of sphere A = 500 V
Potential of sphere B = 1000 V
Distance between their centers = 15 cm
First, calculate the electrical potential energy between the two spheres before they start drifting away:
Potential energy, U = k * q1 * q2 / r
Where U is the potential energy, k is the electrostatic constant, q1 and q2 are the charges on the spheres, and r is the distance between their centers.
Since the spheres are conductive, their charges will be distributed uniformly over their surfaces. The charge on each sphere can be calculated using the formula:
q = C * V
Where q is the charge, C is the capacitance of the sphere, and V is the potential.
The capacitance of a conducting sphere can be calculated using the formula:
C = 4πε0 * R
Where C is the capacitance, R is the radius of the sphere, and ε0 is the vacuum permittivity.
Substituting these formulas into the potential energy equation, we get:
U = (k * (C * V1) * (C * V2)) / r
Now, calculate the initial potential energy between the spheres before they start drifting away.
Next, when the spheres drift away to a distance of 30 cm between their centers, the potential energy becomes zero. This occurs when the spheres have reached a state of equilibrium.
Using the principle of conservation of energy, we can equate the initial potential energy to zero:
U_initial = 0
Using the formula for potential energy, we can set up the equation:
(k * (C * V1) * (C * V2)) / r = 0
Now, solve this equation to find the relationship between the potentials of the spheres.
Once you have the relationship between the potentials, you can determine the potentials of the spheres at various distances. Then using the concept of conservation of charge, you can calculate the charge on each sphere.
Finally, using the concept of conservation of momentum, you can calculate the velocities of the spheres as they drift apart.