Download static electricity - Uplift North Hills Prep

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STATIC ELECTRICITY
Physicists have identified four fundamental forces that account for all phenomena in the universe.
Force
Strength
Distance of action
Description
Strong nuclear
Very strong!
Very, very Short
Holds the nucleus together
Weak nuclear
Very weak
Short
Arise during radioactive decay
Electromagnetic
Very weak
Infinite – but decreases with
the square of distance
Attraction / repulsion of charged
particles
Gravitational
Very, very,
very weak
Infinite – but decreases with
the square of distance
Attraction between masses
With the exception of gravity, all the forces we studied up to now are due, on a molecular level, to interactions
between the electrons of objects – that is, they are caused by the electromagnetic force
normal force
tension
friction force
spring force …
 Charge it is quantized and conserved
ratom ≈ 100000 x rnucleus
mnucleon ≈ 2000 x melectron
Electricity has origin within the atom itself.
•
charge is measured in Coulombs [C]
Electricity has its origin within the atom itself.
• Protons have positive charge (+1e)
• Electrons have negative charge (-1e)
• Neutrons have no charge (0)
e =1.6 x 10-19 C
• Everyday objects - electronically neutral – balance of charge – no net charge.
• Objects can be charged – there can be net charge on an object. How?
The only type of charge that can move around is the negative charge, or electrons. The positive charge stays in the
nuclei. So, we can put a NET CHARGE on different objects in two ways
◊ Add electrons and make the object negatively charged.
◊ Remove electrons and make the object positively charged.
Some materials have atoms that have outer electrons (farthest from nucleus) loosely bound. They can be attracted
and can actually move into an outer orbit of another type of atom. The atom that has lost an electron has a net
charge +e (positive ion). An atom that gains an extra electron has a net charge of – e (negative ion).
This type of charge transfer often occurs when two different materials (different types of atoms) come into
contact.
 Electrical conductors, insulators, semiconductors and superconductors
-
distinction based on their ability to conduct (transfer between materials) electric charge.
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▪ Conductors
A conductor allows electric charge to travel through it easily.
Tap water, human body and metals are generally good conductors.
What makes metals conduct?
• Metals have loosely bound electrons (valence electrons – the electrons in the outermost orbits) –In
metal, atoms are close to each other and valence electrons from each atom get confused and forget
which atom they belong to.
• They now belong to the metal as the whole. Positive ions which are tightly bound and in fixed positions
can only oscillate around their equilibrium positions, form a positive background.
• All the homeless electrons are called “free electrons” or “see of electrons”
• They wander around, keeping ions from falling apart – metallic bond!!
▪ Insulators
•
•
•
•
•
Insulators are materials that impede the flow of electrons from atom to atom.
Insulators have tightly bound electrons – their electrons are not free to move from atom to atom.
This makes insulators poor conductors of heat and electricity.
Examples: air, pure water, plastic, glass, rubber, wood
Only if a very strong electric field is applied, the breakthrough (molecules become ionized resulting in a
flow of freed electrons) could result in destruction of the material.
▪ Semiconductors
`
Materials that can be made to behave sometimes as insulators, sometimes as conductors.
Eg. Silicon, germanium.
▪ ADVANTAGE :As the conductivity of semiconductors can be adjusted by adding certain types of atomic
Impurities in varying concentrations, you can control how much resistance the product will have.
▪ Superconductors
Have zero resistance, infinite conductivity
• Not common! Have to cool to very, very low temperatures.
• Current passes without losing energy, no heat loss.
 Polarization
Charged objects can make neutral objects become polarized. When an object is polarized
• One side has + charge
• The other side has - charge
• The overall charge of the object is zero
▪ Insulators
When bring a charged object near an insulator, electrons are not free to migrate throughout material.
Instead, they redistribute within the atoms/molecules themselves: their “centers of charge” move
When a positively charged object is brought near a neutral insulator, the outermost electrons at each atom will be
drawn to the side facing the positively charged object. As the result, in each atom the
outermost electrons are slightly closer to the positively charged object (and feel an
attractive force), while the positively charged nuclei are slightly farther away (and feel a
repulsive force). But since the outermost electrons are slightly closer to the positively
charged object, and the positively charged nuclei are slightly farther away from the
positively charged object, then the attractive forces will be slightly greater than the
repulsive forces, such that there is a net attraction between the neutral insulator and
positively charged object
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▪ Conductors
In a conductor outer electrons are free to move around within the conductor ("sea of
electrons“).
As the positively charged rod is brought near the conductor, the electrons are attracted
toward the charged rod. Free electrons will move to the side facing towards a positively
charged object. There is a net attraction between the neutral conductor and positively
charged object.
▪ Polar Molecules
Once again a similar effect occurs when a charged object
(positive or negative) is brought near a liquid containing polar
molecules. However, in this case the molecule will reorient
itself such that the negative end will move to the side facing
towards a positively charged object, or away from a negatively
charged object, such that net attraction will result in either
case.
ELECTROSTATIC – ELECTRIC - COULOMB FORCE
 Force between TWO POINT charges q1 and q2 at distance r from each other is proportional to the product of
the amount of the charges on each one, and inversely proportional to the square of the distance between them.
F k
q1q2
r2
Force is a vector (magnitude and direction!)
k  8.99 109 N  m 2 / C 2
. Electric field, E, at a point P distance r away from Q.
● The magnitude of the electric field is defined as the force per unit charge.
E=
F
q
E  
N
C
𝑄
Electric field due to point charge Q 𝐸 = 𝑘
𝑟2
at distance r from it is:
Electric field DOES NOT depend on
the charge placed at that point
● Direction of electric field is direction of the force on
a positive test charge at point P.
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Electric field lines
We use “Electric Field Lines” to visualize el. field.
positive charge
negative charge
El field always point away from + charges, towards – charge
Electric field lines can never cross. If they crossed, that would mean that a charge placed at the intersection, would
be accelerated in TWO directions at once! This is impossible! If two sources are creating electric fields in the same
place, we have to add the two vectors and get a resultant vector representing the NET ELECTRIC FIELD.
Electric field is a vector, so if one has to find electric field duw to more charges one has to do vector sum.
1. Two particles have charges Q and –Q (equal magnitude and opposite sign). For a net force of zero to be exerted on
a third charge it must be placed:
a. midway between Q and –Q
b. on the perpendicular bisector of the line joining Q and –Q, but not on that line itself
c. on the line joining Q and –Q, to the side of Q opposite –Q
d. on the line joining Q and –Q, to the side of –Q opposite Q
e. at none of these places (there is no place)
2. Two point charges, initially 2 cm apart, are moved to a distance of 10 cm apart. By what factor do the resulting
electric and gravitational forces between them change?
a.
5
b. 25
c. 1/5
d. 1/25
3. If the charge and mass are tripled for two identical charges maintained at a constant separation, the electric and
gravitational forces between them will be changed by what factor?
a.
9
b. 2/3
c. 1/9
d. 18
4.
Four charges—A, B, C, and D— are at the corners of a square. Charges A and D, on opposite corners, have equal
charge, whereas both B and C have a charge of 1.0 C. If the force on B is zero, what is the charge on A?
a. –1.0 C
b. –0.20 C
c. –0.35 C
d. –0.71 C
5 Two charges are located on the positive x-axis of a coordinate system. Charge q1 = 2.00 ´ 10–9C, and it is 0.02 m from
the origin. Charge q2 = –3.00 ´ 10–9C, and it is 0.04 m from the origin. What is the electric force exerted by these
two charges on a third charge, q3 = 5.00 ´ 10–9, located at the origin?
a. 2.2 ´ 10–4 N
b. 1.4 ´ 10–4 N
c. 3.1 ´ 10–4 N
d. 8.4 ´ 10–4 N
6. Two charges of –e and +4e are fixed at the positions shown below. At which position along the line XY is the electric
field due to these charges equal to zero?
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7.The diagram below shows two stationary point charges + 2Q and – Q.
At which point is the electric field strength greatest?
A. A
B. B
C. C
D. D
Electrical Energy and Electrical Potential
In the electrical case, a charge placed in electric field will have electric potential energy. How do we know that? If
left on its own it will start accelerating due to electric force acting on it. We say it has electric potential energy
which will be converted into kinetic energy. On the other hand if external force pushes the charge against electric
field, the work done by that force will be stored as potential energy in the charge
Potential energy difference between two points (∆𝑼) is equal to the work done on a charge in order
to move it from one point to the other.
∆U = W
Potential Difference Between Two Points (ΔV = VB – VA)
is equal to the work done per unit positive charge in order to move it from one point to the other.
∆𝑉 =
∆𝑈
𝑞
1 𝑉𝑜𝑙𝑡 =
1 𝐽𝑜𝑢𝑙𝑒
1 𝐶𝑜𝑢𝑙𝑜𝑚𝑏
CHARGE IN UNIFORM FIELD E (constant in magnitude and constant direction)
▪ Work done by electric force on charge q
Electric force on charge q is 𝐹 = 𝑞𝐸.
It moves t
hrough a potential difference, ∆V.
The work done on it by electric force is equal to the decrease in its
electric potential energy which is converted into kinetic energy:
W = ∆U = ½ mv2
q ∆V = ½ mv2
W = Fd = q Ed
▪ Work done by external force on charge q
Electric force on charge q against electric field is 𝐹𝑒𝑥𝑡 = 𝑞𝐸.
It moves through a potential difference, ∆V.
The work done on it by external force against electric field is stored in charge
as the change in electric potential energy U.
W = ∆U
Fext d = q ∆V
qEd = q ∆V ⇒ 𝐸 =
∆𝑉
𝑑
⇒ (E) = NC-1 = Vm-1
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electron-Volt (eV)
How much work is done in moving one electron through a potential difference of one volt?
An electron volt is the amount of energy/work it takes to move an electron through a potential difference of 1 volt.
∆U = W = q ∆V
1 eV = 1.6x10-19 J
The electron volt is not a smaller unit for volts!!! It is a smaller unit for energy.
◊
relationship between uniform electric field E and potential difference between two points distance d from
each other along electric field line:
ΔV = Ed
→ E = ΔV/d and NC-1 = Vm-1
1. An alpha particle (nucleus of helium with 2 protons and 2 neutrons) and an electron are each accelerated from rest
through a potential difference of 2500 volts. Their resultant kinetic energies, in electron volts will be, respectively:
A. none of these
B. 2500, 1.36
C. 10000, 2500
D. 2500, 1250
E. 5000, 2500
2. Two large, parallel, conducting plates are 1.2 mm apart and have equal and opposite charges on their facing
surfaces. An electrostatic force of 6 × 10−13 N acts on an electron placed anywhere between the two plates. What is
the potential difference in volts between the plates?
A. 4500
B. 720
C. 3.76 × 106
D. 2250
4. If 500 J of work are required to carry a 40-C charge from one point to another, the potential difference
between these two points is:
A)
12.5 V
B)
20,000 V
C)
0.08 V
D)
depends on the path
E)
none of these
5. The potential difference between two points is 100 V. If 2 C is transported from one of these points to
the other, the magnitude of the work done is:
A)
200 J
B)
100 J
C)
50 J
D)
100 V
E)
2J
6. A uniform electric field with a magnitude of 500 N/C is directed parallel to the positive x-axis. If the
potential at x = 5 m is 2500 V, what is the potential at x = 2 m? Solutions, detailed in my notes.
a. 1000 V
c. 4000 V
b. 2000 V
d. 4500 V
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REVIEW CIRCUITS
 Resistance of a wire when the temperature is kept constant is:
𝑅=𝜌
𝐿
• length, L
• cross-sectional area, A
• material/resistivity, ρ
𝐴
 OHM’S LAW: Current through resistor is proportional to potential difference
across the resistor and inversely proportional to resistance
of that resistor.
𝐼 =
𝑉
𝑅
𝐼(𝐴)
𝑉(𝑉)
𝑅(𝛺)
 Electric power, P, is the rate at which energy is supplied to or used by a device in which electric energy is
converted into another form such as mechanical energy, thermal energy, or light.
Power dissipated in a resistor:
P=IV
P=
𝑉2
𝑅
= 𝐼2 𝑅
Power of the source = ε I
Electric energy is: 𝐸 = 𝑃 𝑡
𝑠𝑜
𝐸 (𝐽𝑜𝑢𝑙𝑒𝑠) = 𝑃(𝑊𝑎𝑡𝑡𝑠) × 𝑡(𝑠)
𝐸 (𝑘𝑊ℎ) = 𝑃(𝑘𝑊) × 𝑡(ℎ)
 Electromotive force, 𝜺, is the voltage generated by battery (how much energy per unit charge is available for
the circuit including internal resistance)
Resistors in Series
• connected in such a way that all components have the same current through them.
𝑅𝑒𝑞 = 𝑅1 + 𝑅2 + 𝑅3
𝐼=
𝑉
𝑅𝑒𝑞
Resistors in Parallel
• Electric devices connected in parallel are connected to the same two points of an electric circuit, so all
components have the same potential difference across them.
• The current flowing into the point of splitting is equal to the sum of the currents flowing out at that point:
𝐼 = 𝐼1 + 𝐼2 + 𝐼3
𝑎𝑛𝑑 𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑑𝑟𝑜𝑝 𝑖𝑠 𝑒𝑞𝑢𝑎𝑙 𝑎𝑐𝑟𝑜𝑠𝑠 𝑎𝑙𝑙 𝑟𝑒𝑠𝑖𝑠𝑡𝑜𝑟𝑠: 𝐼1 𝑅1 = 𝐼2 𝑅2 = 𝐼3 𝑅3
The greater resistance, the smaller current.
1
𝑅𝑒𝑞
=
1
𝑅1
1
1
+𝑅 +𝑅
𝐼=
2
𝑉
𝑅𝑒𝑞
A device that transforms mechanical energy into electrical energy is called a generator.
A device that transforms electrical energy into mechanical energy is called an electric motor.
A transformer is a device that transforms/change voltage.
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REVIEW PROBLEMS:
1. A circuit is wired with a power supply, a resistor and an ammeter (for measuring current). The ammeter reads a
current of 24 mA (milliAmps). Determine the new current if the voltage of the power supply was ...
a. increased by a factor of 3 and the resistance was held constant.
b. held constant and the resistance was increased by a factor of 2.
c. increased by a factor of 3 and the resistance was decreased by a factor of 2.
d. decreased by a factor of 2 and the resistance was increased by a factor of 2.
2. A 541-Watt toaster is connected to a 120-V household outlet. What is the resistance (in ohms) of the toaster?
3. Consider two appliances that operate at the same voltage. Appliance A has a higher power rating then
appliance B.
a. How does the resistance on A compare with that of B. Is it (1) larger, (2) smaller, or (3) the same?
4. A computer system includes a color monitor with a power requirement of 200 W, whereas a countertop
broiler/toaster oven is rated at 1500 W. What is the resistance of each if both are designed to run at 120 V?
5. Calculate the resistance of an aluminum (ρ = 2.8x10-8 Ωm) wire that is 2.0 m long and of circular cross section
with a diameter of 1.5 mm.
6. The resistance of a certain wire is 10 ohms. What would the resistance of the same wire be if it were twice as long? If it were
twice as thick?
7. A refrigerator operates on average for 10.0 h a day. If the power rating of the refrigerator is 700 W, how much electrical
energy does the refrigerator use in 1 day?
8. An electrical appliance is rated as 2.5 kW, 240 V.
(a) Determine the current needed for it to operate.
(b) Calculate the energy it would consume in 2.0 hours.
9. What is the (total/equivalent)resistance
between points A and B?
10. Find (total/equivalent)resistance resistance for the circuit shown.
11. A 100 Ω, 120 Ω, and 150 Ω resistor are connected to a 9-V battery in
the circuit shown. Which of the three resistors dissipates the most power?
12. What is the current in the battery of the circuit shown below?
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13.
a. Simplify the above circuit so that it consists of one equivalent resistor and
the battery.
b. What is the total current through this circuit?
c. Find the current through each resistor. Find the voltage across each resistor.
d. Find the power of the battery. Find the power dissipated in each resistor.
14. The element of an electric heater has a resistance R when in operation. What is the resistance of a second heater that has
a power output three times as large at the same operating voltage?
15 A heater is required to produce energy, which is 41 of its original value for the same time period. Which of the
following should be done?
A. Double the voltage and half the resistance.
B. Keep the voltage constant but double the resistance.
C. Half the current and double the resistance.
D. Reduce the voltage and resistance each to 41 of their original values.
1. a. Inew = 72 mA
b. Inew = 12 mA c. Inew = 144 mA d. Inew = 6 mA
2. 26.6 Ω
3. smaller
4. 72 W, 9.60 W
5. 32 mΩ
6. 20 2.5 , because twice the diameter gives four times the cross sectional area and one-fourth the resistance
7. E = Pt = (700 W)(36000 s ) = 25.2 x 106 J
or
E = Pt = (0.7 kW)(10.0 h) = 7 kWhours
8. (a) I = P/V = 2500W/240V = 10.4 = 1.0 x 101 A (b) energy = Pt = VIt = (240V)(10.4A)(7200s) = 1.8 x 107 J
𝑅 𝑅
9. 5.2 Ω
10. 𝑅𝑒𝑞 = 1 2 + 𝑅3 + 𝑟
11. 100 Ω
𝑅1 +𝑅2
12. Req = 5 +l5 =20 I = V/R = 0.5 A
13. only the first step :
I = V/Req = 0.5 A
14. R/3
15. D
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REVIEW MAGNETISM
The direction of a magnetic field line is defined as the direction in which the
north pole of a compass points when it is placed in the magnetic field.
Outside the magnet, the field lines emerge from the magnet at its north
pole and enter the magnet at its south pole.
Inside the magnet, there are no isolated poles on which field lines can start
or stop, so magnetic field lines always travel inside the magnet from the
south pole to the north pole to form closed loops.
Magnetic field is measured in Tesla
1 T(Tesla) =
N∙s
C∙m
1. An electric charge experiences a magnetic force when moving in a magnetic field.
Magnetic force acting on a charge q
Magnetic force on a wire carrying current I
in a magnetic Field B: F = qvB sinin a magnetic field B: F = I LB sin
 

q = charge [C]
v = velocity [m/s]
B = magnetic field [Tesla T]
 = angle between v and B
I = current [A]
L = length [m]
B = magnetic field [T]
 = angle between I and B
R-H-R 1: The direction of the magnetic force on a charge/current is given by the right-hand rule 1:
Outstretch fingers in the direction of v (or current I).
Curl fingers as if rotating vector v (I ) into vector B.
Magnetic force on a positive charge (or I) is in
the direction of the thumb.
Magnetic force on a negative charge points in opposite direction.
2. A moving charge produces a magnetic field.
R-H-R 2: The direction of the magnetic field produced by electric current is given by the right-hand rule 2:
If a wire is grasped in the right hand with the thumb in the direction of current flow, the fingers will curl in the
direction of the magnetic field.
= the permeability of free space 4×10-7 T·m/A
𝜇0 𝐼
Magnetic field B around a wire with current I B =
I = current [A]
2𝜋 𝑟
r = distance from the center of the conductor
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Two parallel current carrying wires
attract each other
Two antiparallel current carrying wires
repel each other
Magnetic field of a solenoid with steady current is:
1. uniform from south to north pole inside the solenoid.
(constant in magnitude and direction )
2. not uniform from north to south pole outside of the solenoid.
REVIEW PROBLEMS:
1. The magnetic field of a bar magnet is shown in the figure.
Is the magnet’s north pole at A or B?
2. The direction of the force on a current-carrying wire in an external magnetic field is
a. perpendicular to the current only.
b. perpendicular to the magnetic field only.
c. perpendicular to the current and to the magnetic field.
d. parallel to the current and to the magnetic field.
3. If a proton is released at the equator and falls toward Earth under the influence of gravity, the magnetic force on
the proton will be toward the
a. north.
c. east.
b. south.
d. west.
4. What is the path of an electron moving perpendicular to a uniform magnetic field?
a. a straight line
c. an ellipse
b. a circle
d. a parabola
5. What is the path of an electron moving parallel to a uniform magnetic field?
a. straight line
c. ellipse
b. circle
d. parabola
6. Find the direction of the force on an proton moving
through the magnetic field shown.
7. Find the direction of the force on an electron moving through the magnetic field.
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8. A negative charge is moving through a magnetic field. The direction of motion and the direction
of the force acting on it at one moment are shown. Find the direction of the magnetic field.
9. The direction of the force on a current-carrying wire in an external magnetic field is
a. perpendicular to the current only.
b. perpendicular to the magnetic field only.
c. perpendicular to the current and to the magnetic field.
d. parallel to the current and to the magnetic field.
10. If a proton is released at the equator and falls toward Earth under the influence of gravity, the magnetic force
on the proton will be toward the
a. north.
c. east.
b. south.
d. west.
11. What is the path of an electron moving perpendicular to a uniform magnetic field?
a. a straight line
c. an ellipse
b. a circle
d. a parabola
12. What is the path of an electron moving parallel to a uniform magnetic field?
a. straight line
c. ellipse
b. circle
d. parabola
13. An electron that moves with a speed of 3.0 x 104 m/s perpendicular to a uniform magnetic field of 0.40 T
experiences a force of what magnitude? (e = 1.60 x 10–19 C)
14. An electron moves north at a velocity of 4.5 x 104 m/s and has a force of 7.2 x 10–18 N exerted on it. If the
magnetic field points upward, what is the magnitude of the magnetic field?
15. A 2.0 m wire segment carrying a current of 0.60 A oriented perpendicular to a uniform magnetic field of 0.50 T
experiences a force of what magnitude?
16. A 2.0 m wire segment carrying a current of 0.60 A oriented parallel to a uniform magnetic field of 0.50 T
experiences a force of what magnitude?
17. A current-carrying wire 0.50 m long is positioned perpendicular to a uniform magnetic field. If the current is
10.0 A and there is a resultant force of 3.0 N on the wire due to the interaction of the current and field, what is the
magnetic field strength?
18. A current in a long, straight wire produces a magnetic field. These magnetic field lines
a. go out from the wire to infinity.
c. form circles that pass through the wire.
b. come in from infinity to the wire.
d. form circles that go around the wire.
19. Two long parallel straight wires carry equal currents in opposite directions. At a point midway between the
wires, the magnetic field they produce is:
A) zero
B) non-zero and along a line connecting the wires
C) non-zero and parallel to the wires
D) non-zero and perpendicular to the plane of the two wires
E) none of the above
20. Magnetic field lines inside the solenoid shown are:
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A) clockwise circles as one looks down the axis from the top of the page
B) counterclockwise circles as one looks down the axis from the top of the page
C) toward the top of the page
D) toward the bottom of the page
E) in no direction since B = 0
21. A solenoid is in an upright position on a table. A counterclockwise current of electrons causes the solenoid to
have a(n) ____ magnetic pole at its bottom end. If a compass is placed at the top of the solenoid, the north pole of
the compass would be
a.
north; attracted c.
north; repelled
b.
south; attracted d.
south; repelled
22. A solenoid is in an upright position on a table. A clockwise current of electrons causes the solenoid to have a(n)
____ magnetic pole at its bottom end. If a compass is placed at the top of the solenoid, the north pole of the
compass would be
a.
north; attracted c.
north; repelled
b.
south; attracted d.
south; repelled
23. A long straight wire carries current as shown. Two electrons move with velocities that are parallel and
perpendicular to the current. Find the direction of the magnetic force experienced by each electron.
24. The diagram shows a coil of wire connected to a battery.
Which of the following points is the north pole of the coil?
A. P
B. Q
C. R
D. S
25. A long wire that carries a current I is bent into five loops as shown in the figure.
If the observer could "see" the magnetic field inside this arrangement of loops, how would it appear?
26. The diagram below shows three parallel wires P, Q and R that are equally
spaced.
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The currents in the wires are each of the same magnitude I and are in the directions shown. The resultant force on
wire Q due to the current in wire P and in wire R is
A. perpendicular and into the plane of the paper.
B. perpendicular and out of the plane of the paper.
C. in the plane of the paper to the right.
D. in the plane of the paper to the left.
27. What is the direction of the magnetic field at points P and Q in the plane
of a circular loop carrying a counterclockwise current, as shown?
28. A current-carrying wire is placed perpendicular to the page. Determine the
direction of the electric current from the direction of the magnetic field.
A. Into the page.
B. Out of the page.
C. Clockwise.
D. Counter-clockwise.
E. To the left.
29. Which of the following diagrams represents the magnetic field due to a circular current?
30. A straight long wire carries an electric current to the right. The current is placed in a
uniform magnetic field directed into the page. What is the direction of the magnetic
force on the current?
1. A 2. c
3. c 4. b. 5. a.
6. up, toward the top of the page
7. down, toward the bottom of the page
8. to the left
9. c.
10. c
11. b
12. a
13. 1.9 x 1015 N 14. 1.0 mT
15. 0.60 N 16. 0.0 N
17. 0.60 T
18. d 19. D 20. C 21. d. 22. a
23. A. south
B. west
24. S
25. C 26: D
27. P: out of page
Q: into page
28. B
29. D 30. To the top of the page.