Question

For the purpose of calculating the electric field strength by means of Gauss’s law, determine whether approximate cylindrical symmetry holds in each of the following situations.

Part (a) We have a 3.6-m long copper rod of radius 1 cm, carrying a charge of 1.5 nC distributed uniformly along the rod’s length. We want to calculate the electric field strength at a point 4.9 cm from the rod near its center.
TRUE FALSE
Part (b) We have a 8.9-cm long copper rod of radius 1 cm, carrying a charge of 1.5 nC distributed uniformly along the rod’s length. We want to calculate the electric field strength at a point 4.9 cm from rod near its center.
TRUE FALSE
Part (c) A 1.9-m long wooden rod is glued end-to-end to a 1.9-m long plastic rod, both of radius 1 cm. The combined rod is then painted with an electrically charged paint so that it is covered with a uniform charge density, giving it total charge of 1.5 nC. We want to calculate the electric field strength at a point 4.9 cm from the rod near its center.
TRUE FALSE
Part (d) For the same charged rod as in part (c) we want to calculate the electric field strength at a point 8.9 m from each end of the rod.
TRUE FALSE

Answer 1

For part (a), the approximate cylindrical symmetry holds. For part (b), the approximate cylindrical symmetry does not hold. For part (c), the approximate cylindrical symmetry does not hold. For part (d), the approximate cylindrical symmetry holds.

Step-by-step explanation:

Answer:

In order to determine whether approximate cylindrical symmetry holds for each situation, we need to assess if the distribution of charge is uniform along the length of the rod.

Part (a):

For a 3.6-m long copper rod of radius 1 cm, carrying a charge of 1.5 nC distributed uniformly along the rod’s length, the approximate cylindrical symmetry holds because the charge is evenly distributed. Therefore, the answer is TRUE.

Part (b):

For an 8.9-cm long copper rod of radius 1 cm, carrying a charge of 1.5 nC distributed uniformly along the rod’s length, the approximate cylindrical symmetry does not hold because the length of the rod is too short compared to its radius. Therefore, the answer is FALSE.

Part (c):

A wooden rod and plastic rod glued together and painted with a uniform charge density does not exhibit approximate cylindrical symmetry since it is composed of two different materials and the charge distribution may not be entirely uniform. Therefore, the answer is FALSE.

Part (d):

For the same charged rod as in part (c), the electric field strength at a point 8.9 m from each end of the rod does not depend on the symmetry of the rod. It only depends on the distance from the end of the rod. Therefore, the answer is TRUE.

Answer 2

1. True

2. False

3. True

4. False

Step-by-step explanation:

a) In this case, length of the copper rod is 3.6 m which is much larger than the distance 4.9 cm to the point at which electric field is to be determined. Therefore, yes, cylindrical symmetry holds.

b) In this case, length of the copper rod is 8.9 cm which is of the same order of magnitudes the distance 4.9 cm to the point at which electric field is to be determined. Therefore, no, cylindrical symmetry does not hold.

c) In this case, length of the copper rod is 3.8 m which is much larger than the distance 4.9 cm to the point at which electric field is to be determined. Therefore, yes, cylindrical symmetry holds.

d) In this case, length of the copper rod is 3.6 m which is of the same order of magnitudes the distance 4.9 cm to the point at which electric field is to be determined. Therefore, no, cylindrical symmetry does not hold.