A 10.0-μF capacitor has plates with vacuum between them....

A $10.0-\mu \mathrm{F}$ capacitor has plates with vacuum between them. Each plate carries a charge of magnitude 1000$\mu \mathrm{C}$ . A particle with charge $-3.00 \mu \mathrm{C}$ and mass $2.00 \times 10^{-16} \mathrm{kg}$ is fired from the positive plate toward the negative plate with an initial speed of $2.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Does the particle reach the negative plate? Explain how you can tell. If it does, what is its impact speed? If it does not, what fraction of the way across the capacitor does it travel?

Key Concepts

Capacitor

A capacitor is an electrical device consisting of two conductive plates separated by an insulating medium, which stores energy in the form of an electric field. The amount of energy stored is determined by its capacitance, a measure of the ability to hold charge for a given voltage, and is influenced by physical factors such as plate area and separation distance.

Electric Field in a Capacitor

In a capacitor, the electric field between the plates is typically uniform and is created by the separation of positive and negative charges on the plates. This field is responsible for the force exerted on any charged particle placed within it, and its strength can be calculated based on the charge density and the properties of the medium between the plates.

Electric Potential Difference

The electric potential difference (or voltage) across the capacitor’s plates is the work done per unit charge to move a charge between the plates. This potential difference directly relates to the stored energy in the capacitor and plays a critical role in determining the change in potential energy for a charged particle moving within the field.

Energy Conservation in Electric Fields

The principle of energy conservation states that the total energy in a closed system remains constant. When a charged particle moves through an electric field, its potential energy is converted into kinetic energy (or vice versa) without any loss of total mechanical energy, allowing predictions of motion based on initial speed and potential energy differences.

Work-Energy Principle

The work-energy principle connects the work done by forces on an object to its change in kinetic energy. In an electric field, the work performed by the field on a charged particle equals the change in the particle’s kinetic energy, enabling calculations of acceleration, final velocities, or displacement by equating the work to the difference between initial and final kinetic energies.

Transcript

00:01 Hi, here in this given problem capacitance of the capacitor that is given as 10 .0 microfarad.

00:19 The charge over its plates q is equal to 1000 microcoulomb and charge free to move between the plates that is q is equal to minus 3 .00 microcoulomb.

01:00 Mass of this particle, charge particle which is free to move 2 .00 multiplied by 10 to the power minus 16 kilogram.

01:12 Its initial velocity, initial velocity given to it that is 2 .00 into 10 to the power 6 meter per second.

01:20 So, kinetic energy, initial kinetic energy of the particle or we can say given to the particle e k that is given as half m into v naught square.

01:41 So, plugging in the known values this kinetic energy is half multiplied by half into 10 to the power minus 16 multiplied by 2 into 10 to the power 6 square.

01:57 So, this kinetic energy imparted to this particle comes out to be equal to 4 into 10 to the power minus 4 joules and work required to move this negatively charged particle from positively charged plate, positive plate to negative plate as it will be repelled by the negative plate.

02:35 So, we have to push it towards the negative plate.

02:37 So, work done will be given by q into delta v, potential difference between the plates and potential difference between the plates that is given as q by c for capacitor.

02:51 So, this is 3 into 10 to the power minus 6 taking the magnitude of the charge multiplied by the charge 1000 micro coulomb divided by capacitance which is 10 micro farad...

Please give Ace some feedback