A sphere of unknown mass with a net charge, Q is released from rest in a uniform electric field, E. After it travels a distance,d , it acquires a kinetic energy, K . Find the magnitude of the electric field in terms of K,Q , and .d [Hint: remember the Work/Energy theorem]. The effects of work done by gravity are negligible.
I know the work energy theorem, but how does this relate to charge?
Thank you.
work done=force*d=Eqd
ke gained=work done
K=Eqd
E=K/qd
Well, imagine the situation like this: the sphere starts with no kinetic energy, so all of its initial energy must have come from the electric field. As the sphere moves through the field, it gains some speed and therefore kinetic energy. That means that some of the electric potential energy has been converted into kinetic energy.
Now, according to the work-energy theorem, the work done on an object is equal to the change in its kinetic energy. In this case, the work done by the electric field is equal to the change in the sphere's kinetic energy. Mathematically, we can write:
Work done by electric field = Change in kinetic energy
The work done by the electric field can be calculated as the product of the electric field strength (E) and the distance traveled (d):
Work done by electric field = E * d
And the change in kinetic energy is given as K. Therefore, we have:
E * d = K
Now we can solve for the magnitude of the electric field, E, in terms of K, Q, and d:
E = K / d
Since the mass of the sphere is unknown and the effects of gravity are negligible, we don't need to consider them in this scenario.
The work-energy theorem states that the net work done on an object is equal to the change in its kinetic energy. In this case, the sphere with a net charge Q is released from rest in a uniform electric field E, and it travels a distance d. We want to find the magnitude of the electric field in terms of K, Q, and d.
To apply the work-energy theorem, we need to consider the work done by the electric field on the charged sphere. The work done by the electric field is given by:
W = F * d
Where W is the work done, F is the force exerted by the electric field, and d is the distance traveled by the sphere. In this case, the force exerted by the electric field can be written as:
F = Q * E
Where Q is the charge of the sphere and E is the electric field.
Substituting the expression for force into the work formula, we get:
W = Q * E * d
Since the work done by the electric field results in a change in kinetic energy, we can also express the work as:
W = K - 0
Where K is the final kinetic energy of the sphere and 0 is the initial kinetic energy (which is zero because the sphere is released from rest).
Now we can equate the two expressions for work:
K - 0 = Q * E * d
Simplifying this equation, we find:
K = Q * E * d
Finally, we can solve for the electric field E:
E = K / (Q * d)
So, the magnitude of the electric field in terms of K, Q, and d is K / (Q * d).
The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, we can relate the work done on the sphere with its change in kinetic energy.
When a charged object moves in an electric field, the electric field does work on the charge, given by the equation:
Work = force × distance × cos(θ)
Here, force is the electric force acting on the charge, distance is the distance traveled, and θ is the angle between the force and displacement vectors. The electric force acting on the charge is given by the equation:
Force = Q × E
where Q is the charge on the sphere and E is the electric field.
If the sphere acquires a kinetic energy, K, then the work done by the electric field can be equated to the change in kinetic energy:
Q × E × d × cos(0) = K
Since cos(0) = 1, we can simplify the equation to:
Q × E × d = K
Now, we can solve the equation to find the magnitude of the electric field, E, in terms of K, Q, and d:
E = K / (Q × d)