30: Introduction to Electric Current

INTRO:
To understand the nature of electric current and the conditions under which it exists.
Electric current is defined as the motion of electric charge through a conductor. Conductors are materials that contain movable charged particles. In metals, the most commonly used conductors, such charged particles are electrons. The more electrons that pass through a cross section of a conductor per second, the greater the current. The conventional definition of current is

I=Q_total/Δt

where I is the current in a conductor and Qtotalis the total charge passing through a cross section of the conductor during the time interval Δt.

The motion of free electrons in metals not subjected to an electric field is random: Even though the electrons move fairly rapidly, the net result of such motion is that Q_total=0 (i.e., equal numbers of electrons pass through the cross section in opposite directions). However, when an electric field is imposed, the electrons continue in their random motion, but in addition, they tend to move in the direction of the force applied by the electric field.

In summary, the two conditions for electric current in a material are the presence of movable charged particles in the material and the presence of an electric field.

Quantitatively, the motion of electrons under the influence of an electric field is described by the drift speed, which tends to be much smaller than the speed of the random motion of the electrons. The number of electrons passing through a cross section of a conductor depends on the drift speed (which, in turn, is determined by both the microscopic structure of the material and the electric field) and the cross-sectional area of the conductor.

In this problem, you will be offered several conceptual questions that will help you gain an understanding of electric current in metals.
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PART A:
You are presented with several long cylinders made of different materials. Which of them are likely to be good conductors of electric current?

• copper
• aluminum
• glass
• quartz
• cork
• plywood
• table salt
• gold

SOLUTION:
As stated in the intro, metals are most likely to be good conductors of electric current,
so:
copper, aluminum, and gold
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PART B:
Metals are good conductors of electric current for which of the following reasons?

• They possess high concentrations of protons
• They possess low concentrations of protons
• They possess high concentrations of free electrons
• They possess low concentrations of free electrons

SOLUTION:
The intro states that "In metals, the most commonly used conductors, such charged particles are electrons."
so, the third option: They possess high concentrations of free electrons
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PART C:
Which of the following is the most likely drift speed of the electrons in the filament of a light bulb?

• 10^-8 m/s
• 10^-4 m/s
• 10 m/s
• 10⁴ m/s
• 10⁸ m/s

SOLUTION:
The second option,  10^-4 m/s

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PART D:
You are presented with several wires made of the same conducting material. The radius and drift speed are given for each wire in terms of some unknown units r and v. Rank the wires in order of decreasing electron current.
Rank from most to least electron current. To rank items as equivalent, overlap them.

SOLUTION:
Since the wires are made of the same material, the charge carriers and their densities are the same for all the wires.

Other conditions being equal, the current is proportional to the product of the cross-sectional area of the wire and the drift velocity, that is,

I=n|q|v_dA,

where I is the current, v_d is the drift velocity, A is the cross-sectional area, n is the density of charge carriers, and q is the charge on the carriers.

Therefore:

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PART E:
The drift speed of the electrons in a wire depends strongly on which of the following factors?

• The cross-sectional area of the wire
• The mass of the wire
• The temperature of the wire
• The internal electric field in the wire

SOLUTION:
In the intro, it states "the motion of electrons under the influence of an electric field is described by the drift speed"
so, the final option is correct: The internal electric field in the wire
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PART F:
What quality must the charge density on the surface of a conducting wire possess if an electric field is to act on the negatively charged electrons inside the wire?

The charge density must be... 

• positive
• negative
• nonuniform
• uniform

SOLUTION:
nonuniform

30: Problem 30.6

PART A:
How many conduction electrons are there in a 4.50 mm diameter gold wire that is 20.0 cm long?

SOLUTION:
signature first step...

Givens/ conversions:

Diameter ≡ d = 4.50 mm = 0.0045 m
Length ≡ l = 20.0 cm = 0.200 m

Using Table 30.1...

n_eGOLD = 5.9×10²⁸ m^-3

And we need to determine N_e... 

Equation (30.2) states that N_e = n_e⋅V
We can determine volume by V = A⋅l = (π/4)d²⋅l

∴ N_e = n_e⋅(π/4)d²⋅l

= (5.9×10²⁸ m^-3)(π/4)⋅(0.0045 m)²⋅(0.200 m)

⇒N_e = 1.88×10²³ electrons

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PART B:
How far must the sea of electrons in the wire move to deliver -38.0 nC of charge to an electrode?
Express your answer with the appropriate units.

SOLUTION:
So we know from part A that in 0.2 m, there are 1.877×10²³ electrons ...

each electron has a charge of -1 e, or -1.602×10^-19 C
and that 1 nC = 1×10^-9 C, or 1×10^-9 C/nC = 1×10⁹ C/nC....

this means that the desired charge = -38×10^-9 C

The total charge created by that many electrons is simply equal to the number of electrons multiplied by the charge or each individual electron, or Q_tot = Q_e⋅N_e

The charge per unit length can be determined by dividing the total charge by the length from part A.

Q/l = Q_e⋅N_e/l = (-1.602×10^-19 C)(1.877×10²³)/(0.2 m)
→ Q/l = -1.504×10⁵ C/m
lets call this Q_perLength
​

The total length (L) required to contain a total charge (Q_f) may then be solved for by ... L = Q_f/Q_perLength = l⋅Q_f/(Q_e⋅N_e) 

→ L = (0.2 m)(-38×10^-9 C)/[(-1.602×10^-19 C)(1.877×10²³)]
⇒ L = 2.528×10^-13 m

Which is a really small number... which brings us to the last aspect. Units... I think that the units (which are units of length, of course) are most likely going to be desired in the same units that the length in part A was expressed, so centimeters... 

10² cm/m... so

⇒ L = 2.53×10^-11 cm

25: Charging a Conducting Rod

INTRO:

This problem explores the behavior of charge on conductors. We take as an example a long conducting rod suspended by insulating strings. Assume that the rod is initially electrically neutral. For convenience we will refer to the left end of the rod as end A, and the right end of the rod as end B. In the answer options for this problem, "strongly attracted/repelled" means "attracted/repelled with a force of magnitude similar to that which would exist between two charged balls.

Part A:
A small metal ball is given a negative charge, then brought near (i.e., within about 1/10 the length of the rod) to end A of the rod. What happens to end A of the rod when the ball approaches it closely this first time?

a) It is strongly repelled.
b) It is strongly attracted.
c) It is weakly attracted.
d) It is weakly repelled.
e) It is neither attracted nor repelled.

SOLUTION:
It is stated that the ball is much closer to the end of the rod than the length of the rod. Therefore, if points down the rod several times the distance of approach (but still much closer to end A than end B) are to experience no electric field, the charge on end A of the rod must be comparable in magnitude to the charge on the ball (so that their fields will cancel).

If you can recall the problem from this section, A Test Charge Determines Charge on Insulating and Conducting Balls, and the points made regarding conductors, it can be ascertained that in conductors, the electrons are free to move about. This means that when a charge is brought near to a conductor, the opposite charges all navigate to the point closest the charge and a strong attraction is created
this is true for a conducting rod as well, so the correct answer is option b)

NOTE:
Now consider what happens when the small metal ball is repeatedly given a negative charge and then brought into contact with end A of the rod.

Part B:
After a great many contacts with the charged ball, how is the charge on the rod arranged (when the charged ball is far away)?

a) There is positive charge on end B and negative charge on end A.
b) There is negative charge spread evenly on both ends.
c) There is negative charge on end A with end B remaining neutral.
d) There is positive charge on end A with end B remaining neutral.

SOLUTION: 
in a conductor, no matter how many times it comes in contact with a charge, if it returns to a static situation the charge returns to being neutral. This is satisfied by option b)

Part C:
How does end A of the rod react when the charged ball approaches it after a great many previous contacts with end A? Assume that the phrase "a great many" means that the total charge on the rod dominates any charge movement induced by the near presence of the charged ball.

a) It is strongly repelled.
b) It is strongly attracted.
c) It is weakly attracted.
d) It is weakly repelled.
e) It is neither attracted nor repelled.

SOLUTION:

The answer is a) 
but I can't exactly explain why
I am rather confused from this one 

Part D:
How does end B of the rod react when the charged ball approaches it after a great many previous contacts with end A?

a) It is strongly repelled.
b) It is strongly attracted.
c) It is weakly attracted.
d) It is weakly repelled.
e) It is neither attracted nor repelled.

SOLUTION:
Because the rod is a conductor, the charge is free to distribute itself over the entire rod. It must be distributed so that the internal electric field in the rod is zero, and there is only one distribution that achieves this. There is no memory in this situation: The charge will always distribute itself into the same final result. The rod is symmetric. Therefore, the final distribution of charge must also be symmetric, and hence the same charge must be on end A as on end B.
so the answer is a)

25: A Test Charge Determines Charge on Insulating and Conducting Balls

INTRO: 
When a test charge is brought near a charged object, we know from Coulomb's law that it will experience a net force (either attractive or repulsive, depending on the nature of the object's charge). A test charge may also experience an electric force when brought near a neutral object. Any attraction of a neutral insulator or neutral conductor to a test charge must occur through induced polarization. In an insulator, the electrons are bound to their molecules. Though they cannot move freely throughout the insulator, they can shift slightly, creating a rather weak net attraction to a test charge that is brought close to the insulator's surface. In a conductor, free electrons will accumulate on the surface of the conductor nearest the positive test charge. This will create a strong attractive force if the test charge is placed very close to the conductor's surface.

Consider three plastic balls (A, B, and C), each carrying a uniformly distributed charge equal to either +Q, -Q or zero, and an uncharged copper ball (D). A positive test charge (T) experiences the forces shown in the figure when brought very near to the individual balls. The test charge T is strongly attracted to A, strongly repelled from B, weakly attracted to C, and strongly attracted to D.


Assume throughout this problem that the balls are brought very close together.

Part A: 
What is the nature of the force between balls A and B?

a) strongly attractive
b) strongly repulsive
c) weakly attractive
d) neither attractive nor repulsive

SOLUTION:
We begin by determining the net charges of balls A & B based on the reactions of the test charge near the balls, respectively 
We know that the test charge is positively charged
since there is a strongly attractive force between ball A and the test charge, the nature of the net charge of ball A must be negative
A = -Q
since there is a strongly repulsive force between ball B and the test charge, the nature of the net charge of ball B must be positive
B = +Q

so the nature of the force between a negative ball A and a positive ball B would be strongly attractive, or option a)

Part B:
What is the nature of the force between balls A and C?

a) strongly attractive
b) strongly repulsive
c) weakly attractive
d) neither attractive nor repulsive 

SOLUTION:
We already know the net charge of ball A from the previous part,
A = -Q
& we must also determine the net charge of ball C
In the intro, it is stated that "A test charge may also experience an electric force when brought near a neutral object... In an insulator, the electrons are bound to their molecules... they can shift slightly, creating a rather weak net attraction to a test charge that is brought close to the insulator's surface"
what this means is that C is a neutral insulator, based on its weak net attraction to the test charge. 
so, no matter the charge of the ball brought near to ball C, as long as it has any charge it will result in a weakly attractive force between the two balls, or option c)

Part C:
What is the nature of the force between balls A and D?

a) attractive
b) repulsive
c) neither attractive nor repulsive 

SOLUTION:
We already know the net charge of ball A from part A,
A = -Q
& we must use this to determine the nature of the forces between that ball and this uncharged copper ball D
In the intro it states that "In a conductor, free electrons will accumulate on the surface of the conductor nearest the positive test charge. This will create a strong attractive force if the test charge is placed very close to the conductor's surface."
so when ball A is brought close to ball D, the nature of the surface charge density on the side of ball D that is closest to ball A will be positive
This will result in an attractive force between the balls, or option a)

Part D:
What is the nature of the force between balls D and C?

a) attractive
b) repulsive
c) neither attractive nor repulsive 

SOLUTION:
C is a neutral insulator which will only react to charges brought near to it, as we determined in part c
and D is a neutral copper conductor which will only react to charges brought near to it as well
but since neither ball has a charge, when they are brought near to each other, it will result in neither an attractive nor a repulsive force, or option c)