Physics Electrostatic Potential and Capacitance

If the electric potential at any point (x, y, z)m in space is given by V = 3x² volt. The electric field at the point (1, 0, 3)m will be:

A capacitor is discharging through a resistor R. Consider in time t₁ the energy stored in the capacitor reduces to half of its initial value and in time t₂, the charge stored in the capacitor reduces to half of its initial value and in time t₂ the charge stored reduces to one eight of its initial value. The ratio t₁/t₂ will be

Initially the capacitor is uncharged. What is the steady state charge on it $\left(in\mu C\right)$  after the switch is closed?

An electron moves in the field produced by a charged sphere of radius 10cm along the radius connecting the points separated by 12 and 15 cm from the centre of the sphere. The velocity of the electron changes thereby from 2 * 10⁵ to 2 * 10⁶ m/s. Determine the surface charge (in nC/m²) density of the sphere. (Mass of electron = 9.1 * 10^-31kg)

Two air filled capacitors are connected as shown. The left capacitor is suddenly filled with a dielectric of constant K. The work done by battery after that is $24\mu J.$  Dielectric constant K is:-

Three capacitors C₁ = $2\mu F,C_2=12\mu F$  and C₃ = $12\mu F$ are connected as shown in figure. Find the ratio of the charges on capacitors C₁, C₂ and C₃ respectively:

A capacitor is connected to a 20 V battery through a resistance of 10 $\Omega .$ It is found that the potential difference across the capacitor rises to 2V in 1 $\mu s.$ The capacitance of the capacitor is________ $\mu F.$  

Given  $In\left(\frac{10}{9}\right)=0.105$  

A capacitor of capacitance 3µF is charged to a potential difference of 12V. Another identical capacitor, initially uncharged is filled with a dielectric having dielectric constant K and it is connected to the charged capacitor. The final common potential difference across both capacitors is found to be 2V. Find the value of k.

A proton and an $\alpha$ -particle both initially at rest are accelerated in a region of electric potential difference V and gravity free space. The proton is projected with gained kinetic energy against a uniform constant electric field and comes to momentary rest after travelling a distance $S_0$ . If the $\alpha$ -particle is also projected with gained kinetic energy against same electric field, it will come to momentary rest after travelling through:

Two infinite line charges having linear charge density λ₁ and λ₂ are placed along the x and y axis respectively. The electric field at a point (x₁, y₁) in the first quadrant is inclined at an angle of 60° with the x-axis. Then which of the following relations is correct.