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 keV enters region 1 at time $t=0 .$ That region contains a uniform magnetic field directed into the page, with magnitude 0.010 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 T.The electron goes through a half-circle 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 keV enters region 1 at time $t=0 .$ That region contains a uniform magnetic field directed into the page, with magnitude 0.010 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 T.The electron goes through a half-circle and then leaves region $2 .$ At what time $t$ does it leave?

Key Concepts

Arc Length and Time Calculation in Circular Motion

For particles moving along circular paths, the time to traverse a given arc is determined by the angular displacement and the particle’s angular speed. Recognizing that the path taken might be a fraction of a full circle (as in a semicircular motion) allows us to compute the duration of travel using relationships between arc length, radius, and the period of the circular motion.

Charged Particle Motion in a Magnetic Field

In a uniform magnetic field a charged particle, such as an electron, experiences a Lorentz force that is always perpendicular to its velocity. This force causes the particle to undergo uniform circular motion, where the magnetic force serves as the centripetal force. The radius and period of the circular trajectory are determined by the charge, mass, velocity of the particle, and magnetic field strength.

Relationship Between Kinetic Energy and Velocity

The kinetic energy of a particle is directly related to its speed through the equation KE = 1/2 mv². When a charged particle is accelerated by an electric potential difference, the work done on the particle alters its kinetic energy, thus changing its velocity. This energy-velocity relation is essential for linking potential energy changes to motion parameters in an electromagnetic context.

Motion Under a Uniform Electric Field

When a charged particle traverses a gap across which there is a uniform electric field (created by a constant potential difference), it experiences constant acceleration. The uniform acceleration leads to a linear increase in speed over time, and kinematic equations are employed to calculate the time of transit or the variation of speed along the field direction.

Transcript

00:01 So the question says that there is an electron which enters into the reason one where the magnetic field is into the screen and it makes a semicircle here and then travels a gap of distance 25 cm then enters another magnetic field here then goes a other semicircle here and comes out so we are asked to calculate the total time it takes when it just enters and just exactly out of the reason too okay so there are three phases for this particle one phase here cool one phase here is the particle has to traverse a semicircle here this is one phase then it has to travel this distance then it has to travel another semicircle here so there are three phases for this particle so let us calculate time for all these three so first of all the time for the particle in region one is half of the time period of the particle there now the time period of the particle is what to pi r by velocity because time period is the distance travel by time in velocity right distance travel in one complete rotation divided by velocity so this is equal to pi r by v now using the relation between magnetic force and the centipiddle force let me do this here that is mv square by r is equal to be if e what we can say is that v by r is equal to be e by m so what we can use here is r by v is equal to m by b so pi m by b e right so plug in the values in here so pi is pie mass is for this electron mass is 9 .1 into 10 to the power minus 31 now what is b is b1 here right so b1 is 0 .01 and the charge of electron is 1 .6 into 10 to the power minus 19 so this gives me the time taken for the same circle in the reason 1 and that comes out to be 1 .79 nanoseconds right that is integer to the power minus 9 seconds okay so we are done with the time taken in region 1 let us move to the time taken in region 2 t 2 so for 2 it is a bit of tricky one because the question says that there lies a potential difference of 200 volts across this gap where the electrons is speed increases uniformly so there is an acceleration so to calculate the time period t2, let us use the equation of motion here.

03:30 So let us do that inside first.

03:38 So s is equal to u t plus half 80 square, right? so the distance traveled as is how much? 0 .25 .s is equal to 0 .25 meters...

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