Olivia put a glass of water in the freezer. She left it there for three hours. When she returned, the water had turned to ice. W
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a)
i) Potential for r < a:
ii) Potential for a < r < b:
iii) Potential for r > b:
b) Potential difference between the two cylinders:
c) Electric field between the two cylinders:
Explanation:
a)
Here we want to calculate the potential for r < a.
Before calculating the potential, we have to keep in mind that the electric field outside an infinite wire or an infinite cylinder uniformly charged is
where
is the linear charge density
r is the distance from the wire/surface of the cylinder
By integration, we find an expression for the electric potential at a distance of r:
Inside the cylinder, however, the electric field is zero, because the charge contained by the Gaussian surface is zero:
So the potential where the electric field is zero is constant:
iii) We start by evaluating the potential in the region r > b. Here, the net electric field is zero, because the Gaussian surface of radius r here contains a positive charge density and an equal negative charge density
. Therefore, the net charge is zero, so the electric field is zero.
This means that the electric potential is constant, so we can write:
However, we know that the potential at b is zero, so
ii) The electric field in the region a < r < b instead it is given only by the positive charge distributed over the surface of the inner cylinder of radius a, therefore it is
And so the potential in this region is given by:
(1)
i) Finally, the electric field in the region r < a is zero, because the charge contained in this region is zero (we are inside the surface of the inner cylinder of radius a):
E = 0
This means that the potential in this region remains constant, and it is equal to the potential at the surface of the inner cylinder, so calculated at r = a, which can be calculated by substituting r = a into expression (1):
And so, for r<a,
b)
Here we want to calculate the potential difference between the surface of the inner cylinder and the surface of the outer cylinder.
We have:
- Potential at the surface of the inner cylinder:
- Potential at the surface of the outer cylinder:
Therefore, the potential difference is simply equal to
c)
Here we want to find the magnitude of the electric field between the two cylinders.
The expression for the electric potential between the cylinders is
The electric field is just the derivative of the electric potential:
so we can find it by integrating the expression for the electric potential. We find:
So, this is the expression of the electric field between the two cylinders.
Learn more about electric fields:
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A) Time needed: 6.24 s
B) Time needed: 2.86 s
Explanation:
A)
In this part, we are told that the power if the engine is constant. The power of the engine is given by
where
W is the work done
t is the time
This means that the power of the engine is proportional to the work done, and therefore, to the kinetic energy of the car:
where m is the mass of the car and v its velocity.
SInce power is constant, we can write:
where:
is the time the car needs to accelerates to
is the time the car needs to accelerate to
Therefore, solving for ,
B)
First of all, we have to calculate the acceleration of the car. We can do it using the following equation:
where:
u = 0 is the initial velocity
is the final velocity
t = 1.40 s is the time elapsed
Substituting, we find the acceleration:
In this part, we are told that the force exerted by the engine is constant: according to Newton's second law, acceleration is proportional to the force,
This means that the acceleration is also constant.
Now we want to find how long the car takes to accelerate to a final velocity of
From an initial velocity of
u = 0
Using again the same suvat equation, and using the acceleration we found previously, we find:
Learn more about accelerated motion:
About power:
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