In Fig. 28-32, an electron accelerated from rest through...

In Fig. $28-32$, an electron accelerated from rest through potential difference $V_{1}=1.00 \mathrm{kV}$ enters the gap between two parallel plates having separation $d=20.0 \mathrm{~mm}$ and potential difference $V_{2}=100 \mathrm{~V}$. The lower plate is at the lower potential. Neglect fringing and assume that the electron's velocity vector is perpendicular to the electric field vector between the plates. In unit-vector notation, what uniform magnetic field allows the electron to travel in a straight line in the gap?

In Fig. $28-32$, an electron accelerated from rest through potential difference $V_{1}=1.00 \mathrm{kV}$ enters the gap between two parallel plates having separation $d=20.0 \mathrm{~mm}$ and potential difference
$V_{2}=100 \mathrm{~V}$. The lower plate is at the lower potential. Neglect fringing and assume that the electron's velocity vector is perpendicular to the electric field vector between the plates. In unit-vector notation, what uniform magnetic field allows the electron to travel in a straight line in the gap?

Key Concepts

Electric Field

An electric field is a region where a charged particle experiences a force proportional to the charge. In scenarios like parallel plates with a potential difference, the field is uniform, meaning it has constant magnitude and direction, which simplifies the analysis of particle motion within that region.

Magnetic Field

A magnetic field is a vector field that exerts a force on moving charged particles, with the force being perpendicular to both the particle's velocity and the magnetic field. Uniform magnetic fields, in which this property is consistent throughout the region, can be used to manipulate the trajectory of charged particles in predictable ways.

Lorentz Force

The Lorentz force law describes the total force on a charged particle in the presence of electric and magnetic fields, given by F = q(E + v × B). This principle is fundamental in understanding how electric and magnetic forces combine to influence the motion of particles, as it provides a quantitative framework for calculating net forces.

Force Equilibrium in Charged Particle Motion

When a charged particle moves through overlapping electric and magnetic fields, setting the electric force equal and opposite to the magnetic force can result in a net zero force. This equilibrium condition is essential for achieving straight-line motion, as it cancels the deflection effects of each individual field.

Conversion of Potential Energy to Kinetic Energy

When a charged particle is accelerated through a potential difference, the electrical potential energy is converted into kinetic energy. This conversion is central to determining the particle's speed after acceleration, and it is governed by the principle of energy conservation in electromagnetism.

Transcript

00:01 In this problem we have to find the magnetic field so electron can move on a straight line we know that f net on electron is zero so electric force should be equal to magnetic force we know that electric force is equal to qe and magnetic force is equal to qv cross b here it is given that electron is moving perpendicular to the magnetic field so their angle 90 degree between velocity and magnetic field so we can write electric field is equal to velocity into magnetic field so from here we can write the magnetic field will be equal to electric field divided by velocity now we have to find electric field we know that electric field is equal to potential difference divided by distance between the plates which is d here so we can write electric field is equal to hundred volt divided by d which is 20 mm so i convert it into meter this is electric field now for velocity we know that by energy conservation q delta v that means potential energy should be equal to kinetic energy this is from energy conservation q delta v it is potential energy of electron should be equal to kinetic energy from here we can find the the velocity this is velocity so it will be equal to 2 q delta v by m under root delta v is the potential difference so by putting the value velocity on electric field here in this equation we get the magnet field will be equal to 100 volt divided by 20 into 10 to the power minus 3 meter and divided by velocity which is equal to under root 2 q the charge of electron is 1 .6 into 10 to the power minus 90 into delta v which is 1 into 10 to the power 3 divided by mass which is 9 .1 into 10 to the power minus 31 kg now we will solve this and we get the magnitude of magnetic field is equal to 2 .67 into 10 to the power minus 4 tesla this is the magnitude of magnetic field here now for direction we know that the direction will be in minus k -cap direction so my answer will be equal to 2 .67 into 10 to the power minus 4 k cap.

03:35 Now if we see the diagram for the direction, suppose these are two plates and here electron is moving the straight line and electric field should be in this direction...

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