An electron is accelerated through a uniform electric field

of magnitude 2.5 3 102 N/C with an initial speed of 1.2 3
106 m/s parallel to the electric field, as shown in Figure 10.
(a) Calculate the work done on the electron by the field
when the electron has travelled 2.5 cm in the field.
(b) Calculate the speed of the electron after it has travelled 2.5 cm in the field.
(c) If the direction of the electric field is reversed, how far
will the electron move into the field before coming to
rest?

F = charge in Coulombs * E field strength

a = F/m= force from above / mass of electron
W = work done = F * 2.5*10^-2 meters
(1/2) m Vf^2 = (1/2) m Vi^2 + W

for ( c ), final speed Vf = 0
Wnew = - W = -Fd
and so
F * stopping distance = (1/2) m Vi^2

(a) Well, if an electron is accelerated through an electric field, it means it's getting a little "shock" to move faster. The work done on the electron can be calculated using the formula:

Work = (Force) x (Distance)

In this case, the force is the magnitude of the electric field, which is 2.5 × 10^2 N/C, and the distance is 2.5 cm, which is 0.025 m. So the work done on the electron is:

Work = (2.5 × 10^2 N/C) x (0.025 m)

Now, just crunch the numbers and you'll have the answer!

(b) After the electron has travelled 2.5 cm in the field, it's probably super exhausted! But let's find out its speed anyway. We can use the work-energy theorem, which states that the work done on an object is equal to the change in its kinetic energy:

Work = Change in Kinetic Energy

Since the initial speed of the electron is given as 1.2 × 10^6 m/s parallel to the electric field, we can calculate its initial kinetic energy. Then, we can calculate the final kinetic energy using the work done on the electron calculated in part (a). Finally, we can find the final speed using the final kinetic energy.

(c) Oh dear, the poor electron is going to have a really bad day. If the direction of the electric field is reversed, it means the force it experiences will be in the opposite direction. Eventually, the electron will come to rest when the force opposing its motion equals the force due to the electric field. We can use the equation for force due to an electric field:

Force = Charge x Electric Field

Since the charge of an electron is -1.6 × 10^-19 C and the magnitude of the electric field is 2.5 × 10^2 N/C, we can calculate the distance the electron will move into the field before coming to rest. Just divide the charge by the magnitude of the electric field to find out the distance!

(a) To calculate the work done on the electron by the electric field, we use the formula:

Work = Force * Distance * cos(theta)

Where:
Force = magnitude of the electric field = 2.5 * 10^2 N/C
Distance = 2.5 cm = 2.5 * 10^-2 m
theta = angle between the force and displacement vectors (since they are parallel, theta = 0)

Plugging in these values into the formula:

Work = (2.5 * 10^2 N/C) * (2.5 * 10^-2 m) * cos(0)

Since cos(0) = 1:

Work = (2.5 * 10^2 N/C) * (2.5 * 10^-2 m) * 1

Work = 6.25 J

Therefore, the work done by the field on the electron when it has traveled 2.5 cm is 6.25 J.

(b) The work done on the electron is equal to the change in kinetic energy of the electron. Therefore, we can use the work-energy principle:

Work = Change in Kinetic Energy

Since the electron is initially at rest, the change in kinetic energy is equal to the final kinetic energy when it has traveled 2.5 cm. This can be calculated using the formula:

Kinetic Energy = (1/2) * mass * velocity^2

Where:
mass = mass of the electron = 9.11 * 10^-31 kg
velocity = final velocity (unknown)

Plugging in these values into the formula:

6.25 J = (1/2) * (9.11 * 10^-31 kg) * velocity^2

Solving for velocity:

velocity^2 = (2*6.25 J) / (9.11 * 10^-31 kg)

velocity^2 = 13.69 * 10^30 m^2/s^2

velocity = sqrt(13.69 * 10^30 m^2/s^2)

Therefore, the speed of the electron after it has traveled 2.5 cm in the field is approximately 3.7 * 10^15 m/s.

(c) When the direction of the electric field is reversed, the force on the electron will also be reversed. The force will now oppose the motion of the electron. The electron will eventually come to rest when the force due to the electric field becomes equal in magnitude to the force due to the initial velocity of the particle.

To find the distance the electron moves into the field before coming to rest, we need to find the point where the force due to the electric field equals the force due to the initial velocity.

Force due to the electric field = magnitude of the electric field * charge of the electron

Force due to the initial velocity = mass of the electron * acceleration

Setting these two forces equal:

magnitude of the electric field * charge of the electron = mass of the electron * acceleration

Since the magnitude of the electric field and the charge of the electron are given, we can calculate the acceleration:

acceleration = (2.5 * 10^2 N/C) * (1.6 * 10^-19 C) / (9.11 * 10^-31 kg)

Using the acceleration, we can find the distance traveled before coming to rest:

distance = (1/2) * acceleration * time^2

where time is the time taken to come to rest.

Since we are given the initial velocity, we can calculate the time taken to come to rest:

time = initial velocity / acceleration

Finally, we can substitute the values into the equation for distance:

distance = (1/2) * acceleration * (initial velocity / acceleration)^2

Simplifying the equation:

distance = (1/2) * (initial velocity / acceleration) * (initial velocity / acceleration)

distance = (initial velocity^2) / (2 * acceleration)

Calculating the distance using the given values, we get:

distance = (1.2 * 10^6 m/s)^2 / (2 * acceleration)

distance = (1.44 * 10^12 m^2/s^2) / (2 * acceleration)

Plugging the previously calculated acceleration value:

distance = (1.44 * 10^12 m^2/s^2) / (2 * (2.5 * 10^2 N/C) * (1.6 * 10^-19 C) / (9.11 * 10^-31 kg))

distance = (1.44 * 10^12 m^2/s^2) / (8 * 10^12 N*m^2/C^2)

distance = 0.18 m

Therefore, when the direction of the electric field is reversed, the electron will move approximately 0.18 m into the field before coming to rest.

To calculate the work done on the electron by the electric field, you can use the equation:

Work = force x distance x cosine(theta),

where force is the magnitude of the electric field (2.5 x 10^2 N/C), distance is the distance traveled by the electron (2.5 cm = 0.025 m), and theta is the angle between the force and displacement vectors (which is 0 degrees in this case since the force and displacement vectors are parallel).

(a) Calculate the work done on the electron by the field:
Work = (2.5 x 10^2 N/C)(0.025 m)x cos(0) = 6.25 J

So, the work done on the electron by the field is 6.25 Joules.

To calculate the speed of the electron after it has traveled 2.5 cm in the field, you can use the work-energy theorem:

Work = change in kinetic energy.

In this case, the initial kinetic energy is given as (1/2)mv^2, where m is the mass of the electron and v is its initial speed.

(b) Calculate the speed of the electron after traveling 2.5 cm in the field:
Work = change in kinetic energy
6.25 J = (1/2)m(v^2 - (1.2 x 10^6 m/s)^2)

You need to know the mass of the electron, which is approximately 9.11 x 10^-31 kg, to proceed with the calculation.

Solve the equation for v to find the speed of the electron after it has traveled 2.5 cm in the field.

(c) If the direction of the electric field is reversed, the electron will experience a force in the opposite direction. It will eventually come to rest when the electric force cancels out the force from the initial speed of the electron.

You can use Newton's second law of motion to find the distance the electron will move into the field before coming to rest:

Net force = mass x acceleration.

The acceleration of the electron in this case is given by F/m, where F is the magnitude of the electric field. The net force acting on the electron will be the force due to the electric field minus the force due to the initial velocity of the electron.

Set the net force equal to zero and solve for the distance traveled.

Distance = (initial velocity^2) / (2 x acceleration)

Plug in the given values and solve for the distance to find how far the electron will move into the field before coming to rest.