A uniform electric field of magnitude 640 N / C exists between two parallel plates that are 4.00

A uniform electric field of magnitude 640 N / C exists...

Key Concepts

Uniform Electric Field

A uniform electric field is one in which the field strength and direction remain constant throughout the region of interest. In this context, it provides a constant force on any charged particle placed within it, causing the charged particle to accelerate uniformly. The concept is also fundamental in understanding the behavior of charges between two parallel plates, where the field is generated and can be precisely defined by the voltage difference divided by the distance between the plates.

Force on a Charged Particle in an Electric Field

The force experienced by a charged particle in an electric field is given by F = qE, where q is the charge and E is the electric field strength. This relationship indicates that particles with different charges or masses will experience different accelerations, leading to varying motion behaviors under the same field. This concept is fundamental in analyzing the trajectories of particles such as electrons, protons, or ions under electric fields.

Acceleration in Uniform Electric Fields

Once a charged particle is placed in a uniform electric field, it experiences a constant acceleration determined by Newton's Second Law, a = F/m = qE/m. The uniform acceleration means that standard kinematics equations can be applied to determine quantities such as displacement, time, and velocity. This technique is widely used in solving problems involving motions of particles released from rest in a uniform field.

Equations of Motion in Uniform Acceleration

The kinematics equations for uniformly accelerated motion, such as x = x? + v?t + ½at², are essential in determining the positions and velocities of objects undergoing constant acceleration. In problems involving charged particles, these equations allow us to track the displacement of particles over time when they are subjected to a constant electric force, enabling the determination of the point where two particles meet.

Charge-to-Mass Ratio

The charge-to-mass ratio (q/m) is crucial when analyzing the acceleration of different particles in the same electric field. Since the acceleration is given by a = qE/m, particles with different q/m ratios will accelerate differently even if the electric field and initial conditions are the same. This concept is particularly important when comparing the motion of particles such as electrons, protons, sodium ions, or chloride ions in an electric field.

Transcript

00:01 Okay, so first we can determine the electric force for both particles, since both electron and the proton has a charge 1 .6 times 10 -12 -9 -19 -c -c -cuncc.

00:10 So therefore, the electric force should have same magnitude since the electric field is uniform.

00:16 So therefore, the electric force should be equal to 1 .6 times 10 -2 -9 -19 -cccccccccccccccccccc, which is equal to 1 .024 times 10 -2 -8 -16 newton.

00:26 And then we can determine the acceleration of proton first, which is ap -e -g -o -m -p, and the mass.

00:31 For proton is equal to 1 .67 times 10 to the power negative 27 kilogram.

00:36 So therefore we have the acceleration for proton is about 6 .13 times 10 to the power 10 meter per second square.

00:42 Therefore we can determine the acceleration for electron, let's say ae, which is equal to force divided by the mass of electron.

00:51 And we know the mass of electron is 9 .11 times 10 to the power of negative 31 kilogram.

00:59 And then we'll keep the same force, which is 1 .024 times 10 .000.

01:03 10 to the power of the next 16 newton.

01:06 And this will give us the acceleration is equal to 1 .1 2 times 10 to the power of 14 meter per second square.

01:19 Okay? so when proton and electron pass each other, the distance that was traveled by each particle, if we added it up, it should be equal to the total distance between the plates, which is 4 centimeter.

01:36 So therefore, we can have such equation here, which is, let's say the distance that the proton travel during certain times as sp is equal to bip times t plus 1 half apt square.

01:57 And the distance that was traveled by the electron under certain times equal to vie times t plus 1 half aet squared.

02:11 So v -i -p here is the initial velocity for proton, and v -i here is the initial velocity for the electron.

02:18 And since both of them was at -rest at the beginning, therefore, their initial velocity should be zero.

02:24 So it will just give us asp, which is the distance that was traveled by proton, is equal to 1 -5 -a -t -square.

02:35 And then for the distance -lid -border electrons is equal to 1 -5 -a -e -t -square.

02:40 Okay, and just like what i said, when i pass each other, their total distance should be equal to the distance between the plates, which will have sp plus se is equal to d.

02:58 And d is the distance between the positive plate and the negative plates.

03:02 And i will have 1 half apt square plus 1 half a e t square is equal to d.

03:15 3 times 2 on each side we have apt square plus aet square is equal to 2d.

03:27 Therefore we have t square times ap plus ae is equal to 2d and then we can determine the time square which is equal to 2d over ap plus ae which is equal to 2 times 0 .04 meter because 4 centimeter is equal to 2d is equal to 2d equal to 0 .04 meter.

04:01 Okay.

04:02 And then overly acceleration of the proton plus acceleration of electron, which is 6 .13 times 10 to the power of 10 meter per second square plus 1 .12 times 10 to the power of 14 meter per second square.

04:30 And this will give us t square, which is the time square for each particle when they pass each other.

04:38 Is about 7 .14 times 10 to the power of next 16 second square.

04:49 Okay? and remember question a was asking us the distance that was passed by the proton or the distance from the positive plates.

04:58 Therefore, as p which is the distance that was passed by the positive plates, it's just simply equal to one -half apt square, which is one -half times 6 .13 times 10 to the power of 10 meter per second square and n times t square which is 7 .14 times 10 to the power of next 16 second square okay and this will give us the distance from the positive place is about 2 .18 times 10 to the power of negative 3 meter and let's look at a question b so while we use the same force is because the charge is still 1 .6 times 10 to the power negative 14 coolants since sodium ion is a ion of the positive charge which means that it lost one charge one electron okay and chlorine ion is the negative charge and it means that it was it was gained by one electrons okay so the charge is just one by one electron so therefore the electric force didn't change since the charge didn't change but the mass of the sodium chloride is different since the mass of sodium is 22 .99 atomic mass...

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