(a) An electron is to be accelerated from 3.00 ×10^6 m / s. to 8.00 ×10^6 m / s . Through what p

(a) An electron is to be accelerated from 3.00 ×10^6 m / s....

(a) An electron is to be accelerated from $3.00 \times 10^{6} \mathrm{m} / \mathrm{s}\right.$ to $8.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Through what potential difference must the electron pass to accomplish this? (b) Through what potential difference must the electron pass if it is to be slowed from $8.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ to a halt?

(a) An electron is to be accelerated from $3.00 \times 10^{6} \mathrm{m} / \mathrm{s}\right.$ to $8.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ . Through what potential difference must the electron pass to accomplish this? (b) Through what potential difference must the electron pass if it is to be slowed from
$8.00 \times 10^{6} \mathrm{m} / \mathrm{s}$ to a halt?

Key Concepts

Acceleration and Deceleration in Electric Fields

The process of accelerating or decelerating a charged particle by an electric field relies on the conversion between electric potential energy and kinetic energy. By applying a potential difference, the field can increase a particle’s kinetic energy (acceleration) or reduce it (deceleration), making this concept foundational in understanding particle dynamics in electrical environments.

Electric Potential Difference

Also known as voltage, electric potential difference is the measure of work done per unit charge in moving a charge between two points in an electric field. It is the driving force that accelerates charged particles and is directly related to the amount by which a particle’s kinetic energy increases or decreases.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated as one half the mass times the square of its velocity. When a charged particle such as an electron accelerates or decelerates, its kinetic energy changes, and this change is connected directly to the work done by an electric field across a potential difference.

Conservation of Energy

This principle states that the total energy in an isolated system remains constant. In the context of charged particles moving in an electric field, it means that the work done by the field (through a potential difference) exactly equals the change in kinetic energy of the particle. This concept is crucial when determining how much potential difference is needed to change a particle’s speed.

Transcript

00:01 So in this class string, we have an electron, and we want to do two things about it.

00:07 Part a, we want to accelerate that electron from a speed of 3 .00 times 10 to 6 meter per second to 8 .00 times 10 to 6 meters per second.

00:20 And we want to know what is a potential difference required to accomplish this.

00:25 And part b, we want to know what is, again, what is a potential difference? that will bring it from 8 .00 times 10 to the 6 meters per second to a stop.

00:38 And to do this, we need to use the energy conservation.

00:42 And we know that when electron is speed up or slowdown, what is changing is the kinetic energy of it.

00:50 And because due to the energy conservation, whatever changes of the kinetic energy, it has to be caused by the potential energy.

01:01 The potential energy and kinetic energy will have to add up to be the same.

01:08 So in other words, we know at two points of u1 plus k1 equals v2 plus k2, or actually i should write, it's u1 plus the kinetic energy plus the potential energy at one point is equal to the potential energy plus kinetic energy at the second point.

01:35 So we want to know the potential difference.

01:40 So we want to know delta v equals v1 minus v2 equals u1 minus u2 over q because potential is potential energy divided by q and this is equal to k q minus k1 because this u1 minus u2 equals k2 minus k1 divided by q.

02:03 So now we can plug in all the numbers here.

02:09 We have kinetic energy is one half of mv square, one half an e of v22 minus v1 square over charge of electron.

02:28 Now we're putting some numbers.

02:31 I'll move this down.

02:34 We were putting some numbers.

02:37 One half of the mass of an electron is, you can look it up.

02:43 I think it's 3...

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