An electron accelerated by a potential difference V=3.6 V first enters into a uniform electric

An electron accelerated by a potential difference V=3.6 V...

Key Concepts

Pitch of a Helical Path

The pitch of a helix, defined as the distance along the direction of the magnetic field that a particle travels during one complete cycle of the circular motion, is determined by the product of the parallel velocity component and the period of the circular motion. Calculating the pitch involves understanding the interplay between the electric acceleration processes and the magnetic forces governing the particle's motion.

Electric Potential Energy Conversion

When a charged particle such as an electron is accelerated by a potential difference, its electric potential energy is converted into kinetic energy. This principle allows one to determine the particle’s speed upon exit from an accelerating region using the relation that equates the work done by the electric field to the kinetic energy acquired by the particle.

Time-Varying Electric Fields

A time-dependent electric field, such as one that changes linearly with time, applies a force on a charged particle that varies with time. This introduces an additional acceleration component in the direction of the field. Analyzing this effect involves integrating the acceleration over the time that the particle is within the field, which results in a velocity component perpendicular to the original direction of motion.

Lorentz Force in Magnetic Fields

In the presence of a magnetic field, a charged particle experiences a Lorentz force that is perpendicular to both its instantaneous velocity and the magnetic field direction. This force causes the particle to undergo circular motion in the plane perpendicular to the magnetic field, while any velocity component parallel to the field remains unaffected, setting the stage for helical motion.

Helical Motion of Charged Particles

Helical motion occurs when a charged particle moves in a magnetic field with both parallel and perpendicular components to the field. The perpendicular component leads to circular motion due to the Lorentz force, while the parallel component results in a uniform motion along the direction of the field. The combination of these motions yields a helical trajectory.

Transcript

00:01 Hello, so in this question we have given a parallel plate capacitor of length 6 cm.

00:06 Okay, and this is our positive plate, let's say this is our negative plate.

00:12 And direction of elective field will be like this.

00:15 Direction of electric field will be like this.

00:19 Now a charge particle enters into the parallel plate capacitor which is accelerated with a potential difference of 3 .6 volt.

00:29 4 charge particle, okay, when it is accelerated with v potential difference, then its energy will be equal to e into v, where e is the charge and v the potential difference applied.

00:41 This should be equal to its kinetic energy, that is half m, p, square, okay? so, value of v will be 2ev by m into under root.

00:54 Now charge particle, that is, electron, is given.

00:57 So electron enters into parallel plate capacity.

01:00 With velocity v okay and due to this elective field force acting on the charge particle will be f is equal to q into v sorry q into e where is elective field q is a chart and f equal to m a so m acceleration will be q e by m and this acceleration will act in upward direction okay because force acting on electron will be opposite direction of elective field and also the elective field is variable whose value is equal to a into t where as a positive constant and value is given 3200 3200 volt per meter per second hold per meter per second this is value a okay now what will happen as the charge moves inside capacitor a force will act on a vertical upward direction due to this the charge particle will change its path and moves like in projectile motion okay and comes out from parallel plate capacity let's say some at some angle theta okay so first of all in x direction in x direction there will be no acceleration there will be no acceleration due to this if we consider or we find out value of v that is velocity in x direction will be same as initial velocity okay which is equal to v.

02:31 Now in y direction there is exhalation.

02:34 So velocity in y direction is to find out also the elective field is variable.

02:38 So acceleration in y direction is equal to e into e divided by m, where e into e by m and value of elective field is a into t.

02:53 And acceleration can be written as d -y by d -t, sorry, d -u -y, velocity in y direction.

03:00 So acceleration can be written d, vy by dt is equal to e by m into at.

03:11 So on rearranging this, this will become vy, d, vy is equal to e by m into atdt.

03:24 Okay, now time taken by this charge particle 2 comes out from parallel plate capacitor whose length is given 6.

03:35 Centimeter okay so as there is no acceleration in x direction the velocity is constant always in x direction so particle have to table a total length of six centimeter in x direction therefore the time taken from here we can find out time to cross plate time to cross plate will be equal to simply distance upon velocity okay which is equal to distance that is l by v this is the time taken the particle to cross the plate capacitor okay so similarly in y direction we can find out the velocity after time zero to l by v okay so our y value for velocity in y direction is too big comes out here e m a are constant and integration of t d t will be t square by two t square by two whose limits are zero to l by v okay so this this will become e -a -y -m -l -square upon v -square into 2.

04:50 And in this equation, it can be rearranged like this.

04:55 E -a -l -square, here mv -square.

05:01 Let us multiply this equation by 2 and divide by 2.

05:06 So it will become half m -v -square into 2 and 2.

05:10 It will be 4.

05:12 Now half mv square equal to e into v where v is potential and this is velocity okay.

05:19 So here this becomes half mb square e into v and e and e v cancel out.

05:27 So it will become a l square upon 4 v where v is potential and its value is given.

05:35 Therefore velocity in y direction is comes out a l square upon four v where v is potential and its value is given.

05:41 L square upon 4v this is our vertical velocity and horizontal velocity v x is equal to v okay now after this the charge particle is moving like this with angle theta let's say this is angle of deviation theta and now this interest into magnetic field b whose direction is same as electric field that is vertical downward direction and this is the velocity now due to magnetic field this particle charge particle that is elephant starts moving in circular motion due to magnetic force which is given by f is equal to qv cross b okay that is v cross b okay that is v cross b and when it is entered to a magnetic field it starts moving in circular motion so for circular motion necessary centripetal force that is mv square by r is presented by magnetic force okay so from here we can find out value of r that is m a m .v by bq.

06:45 M .v...

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