Consider a uniformly charged ring in the xy plane, centered
at the origin. The ring has radius a and positive charge q distributed evenly along its circumference.
PartA
What is the direction of the electric fieldat any point on the z axis?
parallel to the x axis | |
parallel to the y axis | |
parallel to the z axis | |
in a circle parallel to the xy plane |
PartB
What is the magnitude of the electric fieldalong the positive z axis?
Use k in your answer, where .
E(z) = | ||
PartC
Imagine a small metal ball of mass m and negative charge -q0. The ball is released from rest at the point(0,0,d) and constrained to movealong thez axis, with no damping. If 0<d≤a, what will be theball's subsequent trajectory?
repelled from the origin | |
attracted toward the origin and coming to rest | |
oscillating along the z axis between z=d and z=-d | |
circling around the z axis at z=d |
The concept used to solve this problem is electric field.
Initially calculate the direction of electric field at any point on the z-axis. After that calculate the magnitude of the electric field along positive z axis. Finally, calculate the ball’s subsequent trajectory.
The expression for electric field is,
Here, is an electrostatic constant, is the vacuum permittivity, is the charge, and is the distance between the charge and point of observation of electric field.
The magnitude of force on a charge in electric field is,
Here, is the charge on the particle in the electric field, and is the electric field.
The projection of any vector along the axis is,
Here, is the magnitude of vector , and is the angle the vector makes counterclockwise from the axis.
The Newton’s second law can be expressed as,
Here, is the vector sum of all the forces acting on the object, is the mass, and is the acceleration of object.
The acceleration along axis is,
Here, is the positions coordinate of the object, and is the time.
The equation of motion of a simple harmonic oscillation along axis is,
Here, is the angular frequency.
In a right-angled triangle, the cosine function is,
Pythagoras theorem gives the hypotenuse of right angled triangle as,
(A)
Draw the diagram of the electric field due to a charge on the uniformly charged ring with a radius and positive charge distributed evenly along its circumference. The magnitude of electric field is . The position coordinate of any arbitrary point on axis is . The angle of electric field makes with axis in counterclockwise direction is . The tangent to a circle is perpendicular to the radius of the circle.
Substitute for , and for in the cosine function equation and solve for .
…… (1)
Substitute for , and for in the equation to get the electric field along axis.
…… (2)
Here, is the electric field along axis.
Use the electric field equation.
Substitute for in the above equation and solve for the magnitude of the electric field due to the uniformly charged ring.
Integrate the equation (2) to solve for the magnitude of electric field due to complete ring.
Here, is the electric field along direction.
Substitute for from equation (1), and for from the equation (3) in the above equation .
(B)
Use the electrostatic force equation.
Here, is the charge on the ball, and is the magnitude of electric field along axis.
Substitute for in the above equation from the part (1).
…… (4)
Substitute for to find the force at any end point of the oscillations in the equation (4).
.
Use the approximation as in the above equation.
Therefore, force at any point along the axis can be approximated from the above result. Substitute for in the above expression to solve for the force along axis.
Use the Newton’s second law to solve for the acceleration of the ball.
Here, is the force acting on the ball, is the mass of the ball, and is the acceleration of the ball.
Substitute for , and for in the above equation and solve for the equation of motion.
Compare this equation of motion with equation of motion of the simple harmonic motion to solve for the angular frequency.
Ans:
The direction of electric field at any point on the z axis is parallel to the z axis.
Consider a uniformly charged ring in the xy plane,centered at the origin. The ring has radius...
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