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Ch 18 Electric Charge and Electric Field
,
,
Coulomb’s Law: The magnitude of the force is
|
||
|
; same sign charges repel, opposites attract.
The electric field is the electric force per unit charge at a point in space: ⃗
. Units: [ ]
The electric field of a positive (negative) point charge points away from (toward) the charge:
, and ⃗
Superposition of forces and fields:
⃗
⃗
| |⁄
.
⃗
Ch 19 Electric Potential and Electric Field
Because electric forces are conservative, we can define the change in electric potential energy by
,
where is the work done by all electric forces, and apply the principle of conservation of mechanical energy:
, or
. The change in electric potential is the change in potential energy per
unit charge:
. Positive charges accelerate in the direction of decreasing potential, while
negative charges accelerate in the direction of increasing potential. Units: [ ]
A useful unit in atomic physics is the electron-volt (
):
The component of ⃗ in any direction is the negative of the rate at which the electric potential changes with
distance in that direction. For a uniform field,
and
. Units: [ ]
Electric potential for a point charge:
where is chosen to be zero at infinity.
Electric potential energy for two point charges separated by a distance :
, also zero at infinity.
Superposition:
The total electric potential due to two or more charges is the sum of
the potentials due to the separate charges.
For a sphere of radius
and charge , we have at the surface
Definition of capacitance :
. Units: [ ]
and
Parallel-Plate Capacitor: If the area of the
plates is , their separation is , and the charge on each plate is , then the potential is
capacitance is
, and the electric field is
.
, the
Dielectrics. Filling a capacitor with a (non-conducting) dielectric material decreases the electric field and
potential difference between the plates. If the dielectric constant is , then
where and
are the values without the dielectric. It is assumed that the charge remains the same. Capacitance of a
Parallel-plate capacitor filled with a dielectric.
Capacitors in series:
, Capacitors in parallel:
The potential energy of a charged capacitor:
( )
( )
Ch 20 Electric Current, Resistance and Ohm’s Law
Definition of electric current, :
Units: [ ]
When voltage is placed across an object to make current flow, the object’s resistance is defined by
Units: [ ]
Ohm’s Law:
and is a constant.
If the object has length
and cross-sectional area
Units: [ ]
Variation with temperature:
reference temperature (often room temperature
the material. Units: If [ ]
, then [ ]
, then the resistivity
), and
.
is defined by
⁄
⁄
, where is the temperature, is the
is the temperature coefficient of resistivity of
The electrical power provided by a source of EMF, or heat dissipating power of a resistor is
Ch 21 Circuits and DC Instruments
Resistors in Series:
(For example, see internal resistance of a battery.)
Resistors in Parallel:
When the battery’s internal resistance is taken into account, the terminal voltage is
.
Kirchhoff’s Junction Rule: The algebraic sum of all the currents meeting at (flowing into) any junction in a
circuit must equal zero. This guarantees that electric charge is conserved. I.e.,
Note that,
if some currents are assumed to flow into the junction, while others are assumed to flow out of the junction,
we have, e.g.,
Kirchhoff’s Loop Rule: The algebraic sum of all potential differences around any closed loop in a circuit is zero.
This includes battery EMFs, and guarantees that the potential is well-defined at any point. I.e.,
Ch 22 Magnetism
Part 22a Magnetic Forces and Magnetic Fields
⃗ . The direction of the magnetic force is given by the Right Hand Rule.
Magnetic force on a particle.
The magnitude is given by
, where is the particle charge, is its speed, is the magnetic field
magnitude, and is the angle between and ⃗ . Units: [ ]
.
If a particle moves with velocity perpendicular to a uniform magnetic field ⃗ , it will undergo uniform circular
motion described by the relation | |
, where is the charge, m is the mass, and is the path
radius.
Crossed fields: For a balance between electric and magnetic forces acting on particle moving with speed , we
must have
. For the EMF in the Hall effect, this leads to
, where is the width of the
conductor.
Part 22b Magnetism and Electric Currents
Magnetic force on a current-carrying wire. If a straight wire segment of length carries current in a uniform
⃗
⃗ ⃗ . The direction of the magnetic force is given by
magnetic field ⃗ , it will receive a force
the Right Hand Rule. The magnitude is given by
, where is the current, is the wire length, is
the magnetic field magnitude, and is the angle between the current and ⃗ .
The torque on a planar rectangular loop of wire, carrying current in the presence of a uniform magnetic field
⃗ is
, where and are the lengths of the rectangle’s sides, and is the angle between the
field and the normal to the loop, using the right-hand rule. If the loop has turns, the torque is
, where is the area of the loop.
Vacuum permeability constant:
.
Magnetic field outside/inside a long straight wire with current : If
from the wire center, then outside the wire (
) the field is
Magnetic field at the center of a single loop of current: If
is the wire radius, and is the distance
.
is the loop radius, then the field is
Magnetic field of a solenoid: Outside the solenoid this is very weak. Inside it is
number of turns per unit length in the solenoid.
.
, where
is the
Force between two parallel currents. To find the force on a current-carrying wire due to a second currentcarrying wire, first find the field due to the second wire at the site of the first wire. Then find the force on the
first wire due to that field. If two parallel wires have currents
and , the force per unit length on a wire
segment of length , due to the other wire, is
, where is the distance between the
wires. Parallel currents attract each other, and antiparallel currents repel each other.
The Ampere. The official definition of the ampere is: One ampere of current through each of two parallel
conductors of infinite length, separated by one meter in empty space free of other magnetic fields, causes a
force of exactly 2×10−7 N/m on each conductor.