In Fig. 28-40, an electron with an initial kinetic energy of 4.0 keV enters region 1 at time t=0

In Fig. 28-40, an electron with an initial kinetic energy of...

In Fig. $28-40$, an electron with an initial kinetic energy of $4.0 \mathrm{keV}$ enters region 1 at time $t=0$. That region contains a uniform magnetic field directed into the page, with magnitude $0.010 \mathrm{~T}$. The electron goes through a half-circle and then exits region 1 , headed toward region 2 across a gap of $25.0 \mathrm{~cm}$. There is an electric potential difference $\Delta V=2000 \mathrm{~V}$ across the gap, with a polarity such that the electron's speed increases uniformly as it traverses the gap. Region 2 contains a uniform magnetic field directed out of the page, with magnitude $0.020 \mathrm{~T}$. The electron goes through a halfcircle and then leaves region 2. At what time $t$ does it leave?

In Fig. $28-40$, an electron with an initial kinetic energy of $4.0 \mathrm{keV}$ enters region 1 at time $t=0$. That region contains a uniform magnetic field directed into the page, with magnitude $0.010 \mathrm{~T}$. The electron goes through a half-circle and then exits region 1 , headed toward region 2 across a gap of $25.0 \mathrm{~cm}$. There is an electric potential difference $\Delta V=2000 \mathrm{~V}$ across the gap, with a polarity such that the electron's speed increases uniformly as it traverses
the gap. Region 2 contains a uniform magnetic field directed out of the page, with magnitude $0.020 \mathrm{~T}$. The electron goes through a halfcircle and then leaves region 2. At what time $t$ does it leave?

Key Concepts

Lorentz Force

The Lorentz force is the fundamental principle governing the motion of a charged particle in electromagnetic fields. It states that a particle with charge q moving with velocity v in the presence of electric field E and magnetic field B experiences a force given by F = q(E + v × B). This force accounts for both the deflection in magnetic fields and the acceleration in electric fields, forming the basis of analyzing the particle's trajectory.

Circular Motion in Magnetic Fields

When a charged particle moves perpendicular to a uniform magnetic field, it undergoes uniform circular motion due to the constant perpendicular Lorentz force acting as the centripetal force. The radius of this motion is found using r = mv/(qB) and the period of the motion is T = 2?m/(qB). These relationships are key to understanding how the particle curves through regions of magnetic influence.

Energy Gain from Electric Potential Difference

A charged particle gains or loses kinetic energy when it moves across a potential difference, according to the work-energy theorem. The change in kinetic energy is given by the product of the charge and the difference in electric potential, ?KE = q?V. This concept explains the increase in the particle’s speed when it passes through regions with an applied electric potential difference.

Time-of-Flight in Composite Trajectories

Analyzing the overall time taken for a charged particle to complete a trajectory that involves different regions (each with either magnetic or electric fields) requires breaking down the path into segments. The time spent in each segment can be determined by the relevant dynamics (circular motion in magnetic fields or linear acceleration in electric fields) and then combined to give the total time-of-flight.

Cyclotron Frequency

The cyclotron frequency, defined as ? = qB/m, represents the angular frequency at which a charged particle orbits in a magnetic field. It is crucial for calculating the duration of specific portions of the motion, such as half-circular or full-circular orbits, and thus plays an important role in timing the particle’s exit from regions with magnetic fields.

Transcript

00:01 In this question, we have electron of initial kinetic energy for keb entering region 1 of uniform magnetic field.

00:13 Case directly into the page and b1, b1 has magnitude 0 .01 tesla.

00:26 Then it enters, it travels in a circular path, right? then exits and then move to through the gap.

00:35 In this gap, there is a potential difference of 2 ,000 volts.

00:39 So meaning here is negative, here is plus 2 ,000, 2 kilowatts.

00:46 Then the electron is accelerated through this gap of 25 cm.

00:53 And then it travels into region 2, perform in the circular motion, and then it exits the region 2.

01:03 Okay, and b2, the magnetic field in b2, region b2 is 0 .02 tesla.

01:13 So we want to find out the time taken for the electron to start from here and exit here.

01:20 Okay, so to do this question, okay, the motion of the, we recognize that the motion of the electron, there are three phases.

01:33 Okay, so phase one.

01:37 Is moving circular motion in in region 1.

01:52 Then phase 2 is accelerate through the tabby of 2 ,000 volts.

02:03 And then phase 3 is circular motion in region 2.

02:14 So then we'll be using the formula, t equals to 2 pi m over bq.

02:25 Phase one and three phases one and three but this t is for the full circle so t1 which is be t over two or region one okay t1 2 and then p3 would be t3 over 2 okay and then for t2 t2 is during the acceleration so let's just to recap.

03:02 So the time, the total time is going to be t1 plus t2 plus t3.

03:08 And the t1, t2 and t3 refers to the time for each phase.

03:14 So we found t1, we found the expression for t1 and t3 already.

03:19 Now we need to find t2.

03:21 So during phase two, the electron is undergoing, constant acceleration motion so we need to find the acceleration of the electron okay and the initial speed of the electron so we'll be using k equals to half mv square to find the initial speed so it goes to 2k over m square so just substitute the numbers, k2 times 4 kb...

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