Question

My Notes Of the charge Q initially on a tiny sphere, a portion q is to be transferred to a second, nearby sphere. Both spheres can be treated as particles and are fixed with a certain separation. For what value of q/Q will the electrostatic force between the two spheres be maximized?

Answer 1

`(q)/(Q)=0.5` m

Step-by-step explanation:

Given:

Initial charge on the tiny sphere=Q

Charge transferred from tiny sphere to another sphere=q

According to coulombs law the electrostatic force between the two spheres is given by

`F=(k(Q-q)q)/(r^2)`

Where r is the fixed distance between them.

For getting maximum force we will differentiate the force with respect to q

Hence we calculated the the ratio for maximum force between them

Answer 2

the electric force between the spheres is maximum when,

`\rm \frac qQ=\frac 12.`

Step-by-step explanation:

Initally, the charge on first sphere = Q.

Now, a portion of q charge is transferred to the second sphere,therefore,

• The charge acquired by the second sphere,
  `\rm q_2` = q.

• The charge remained on the first sphere,
  `\rm q_1` = Q-q.

Let the two spheres are separated by distance
`\rm a`.

According to the Coulomb's law, the electrostatic force of interaction between the two static point charges
`\rm q_1` and
`\rm q_2`, separated by a distance
`\rm r` is given by

`\rm F = (kq_1q_2)/(r^2).`

where,

k is the Coulomb's constant.

It is given that both the spheres can be treated as particles and are fixed with a certain separation.

Therefore, the electrostatic force of interaction between the two spheres is given as:

`\rm F = (k(Q-q)q)/(a^2)=(k(Qq-q^2))/(a^2.)`

The electrostatic force between the two spheres is extremum for the value of
`\rm q=q_o`, when
`\rm\left ( (dF)/(dq)\right )_(q=q_o)=0.`

`\rm \left ( (dF)/(dq)\right )_(q=q_o)=\left [ (d)/(dq)\left ((k(Qq-q^2))/(a^2) \right ) \right ]_(q=q_o)\\\\=\left [ (k)/(a^2)(d)/(dq)\left (Qq-q^2 \right ) \right ]_(q=q_o)\\=(k)/(a^2)(Q-2q_o).`

For,
`\rm\left ( (dF)/(dq)\right )_(q=q_o)=0,`

`\rm (k)/(a^2)(Q-2q_o)=0\\\\\Rightarrow Q-2q_o=0\\q_o=\frac Q2.`

The electrostatic force is maximum when,

`\rm\left ( (d^2F)/(dq^2)\right )_(q=q_o)<0.`

`\rm\left ( (d^2F)/(dq^2)\right )_(q=q_o)=\left ( (d)/(dq)\left ((dF)/(dq)\right )\right )_(q=q_o)\\\\=\left ( (d)/(dq)\left ((k)/(a^2)(Q-2q)\right )_(q=q_o)\\\\=(k)/(a^2)(-2).\\`

`\rm \text{Since, k and a are positive constants, therefore, }\\\left ( (d^2F)/(dq^2)\right )_(q=q_o)<0`

Thus, the electric force between the spheres is maximum when,

`\rm q=q_o = \frac Q2\\\\ i.e.,\ \frac qQ=\frac 12.`