Basic Quantities

• Force $f$: Force can be defined by Newton's second law:
  
  $$f=ma,\;\;\;\;\;\;\;[newton]=\frac{[kilogram][meter]}{[second]^2}$$
  
  
  The force of 1 newton will cause a mass of 1 kilogram to accelerate at 1 meter per second per second.

  Comparison:

  • The gravitational force between masses $m_1$ and $m_2$ is
    
    $$f=G\frac{m_1m_2}{r^2}$$
    
    
    In particular, on surface of Earth, the ``weight'' of a mass $m$ is
    
    $$f=G\frac{Mm}{r^2}=gm$$
    
    
    where $g=GM/r^2=9.8\;\mbox{meter}/\mbox{second}^2$ is the gravitational acceleration that measures the intensity of Earth's gravitational field.
  • The electric force is
    
    $$f=k\frac{q_1q_2}{r^2}=Eq_2$$
    
    
    where $E=kq_1/r^2$ is the intensity of the electric field caused by charge $q_1$, and $f$ is the force this field asserts on charge $q_2$.

• Energy $W$: (a conservative quantity, can be neither created nor destroyed)

  Potential Energy: When an object with weight of 1 newton is raised by 1 meter, it receives 1 joule of potential energy.
  

  $$W=fl,\;\;\;\;\;\;[joule]=[newton][meter] =\frac{[kilogram][meter]^2}{[second]^2}$$
  
  
  In gravitational field, the potential energy is $W=fl=mg(h_2-h_1)$, while in electric field, it is $W=fl=Eql$, where the field intensity is $E=f/q$.

  Kinetic Energy: A mass of 1 kilogram moving with a velocity $u$ of 1 meter per second possesses 1/2 joule of kinetic energy.
  

  $$W=mu^2/2,\;\;\;\;\;[joule]=\frac{[kilogram][meter]^2}{[second]^2}$$
  
  
  Another unit for energy is calorie: $1 \; \mbox{cal}=4.2\;\mbox{joules}$

• Power $P$: The rate of energy transformation. The transformation 1 joule of energy in 1 second represents a power of 1 watt. The instantaneous and average power are defined by:
  
  $$p=dw/dt,\;\;\;\;\;\;P=W/T\;\;\;\;\;[watt]=\frac{[joule]}{[second]}$$
  
  
  Another unit for power is horse power: $1 \; \mbox{hp}=746 \; \mbox{watts}$

  Energy can also be measured by kilowatt-hours (kWh) $1000 \times 3600=3.6 \times 10^6$ joules.

• Charge: The charge of $6.24 \times 10^{18}$ electrons is a coulomb, i.e., the charge each electron carries is $1/ (6.24 \times 10^{18})=1.602 \times 10^{-19}$ coulomb. Charge is conservative.

• Current: The rate of flow of positive charges:
  
  $$i=dq/dt,\;\;\;\;\; q=\int i \; dt, \;\;\;\;\; [\mbox{ampere}]=\frac{[coulomb]}{[second]}$$
  
  

• Voltage v: The energy required to move a unit charge $q=1$ from one electric potential $u_1$ to another $u_2$. The difference between the two potentials or voltage is $v=u_2-u_1$. When a charge of 1 coulomb is moved through a voltage of 1 volt, it receives or delivers an energy of 1 joule
  
  $$v=\frac{w}{q}=\frac{Eql}{q}=El,\;\;\;\;[volt]=\frac{[joule]}{[coulomb]}$$
  
  

  Comparison:

  • The energy needed to move a charge $q$ from point 1 with potential $u_1$ to point 2 with potential $u_2$ is $w=qv=q(u_2-u_1)$, and $v=u_1-u_2$ is the potential difference between the two points.
  • The energy needed to move a mass $m$ from height $h_1$ to height $h_2$ is $w=fl=mg(h_2-h_1)$, and $gl=g(h_2-h_1) =gh_2-gh_1$ is the potential difference between the two heights.

• Electric Field: a force field defined as:
  
  $$E=\frac{dv}{dl}=\frac{u_2-u_1}{l}$$
  
  

  Comparison:

  • Electric field $E=(u_2-u_1)/l$ is the electric potential difference per unit distance.
  • Gravitational field $g=(gh_2-gh_1)/l$ is the gravitational potential difference per unit distance.

  But as
  

  $$E=\frac{v}{l}=\frac{w}{ql}=\frac{fl}{ql}=\frac{f}{q}$$
  
  
  we get the alternative definition of electric field
  
  $$E=f/q, \;\;\;\;\; f=q\;E$$
  
  

  Comparison:

  • Electric field $E=f/q$ is electric force per unit charge.
  • Gravitational field $g=f/m$ is gravitational force per unit mass.

  

• Electrical power and energy:
  
  $$p=\frac{dw}{dt}=\frac{dw}{dq} \; \frac{dq}{dt}=vi, \;\;\;\;\... ...coulomb]}\frac{[coulomb]}{[second]} =\frac{[joule]}{[second]}$$
  
  
  
  
  $$w=\int p dt = \int vi dt,\;\;\;\; [joule]=[watt][second]=[volt][ampere][second]$$
  
  

• Magnetic Flux: The intensity of magnetic effect (lines per unit area in magnetic field or flux) is measured by magnetic flux density $B$ in tesla, and the magnetic flux $\phi$ within an area is
  
  $$\phi=\int B \cdot dA=\vert B\vert\;\vert A\vert cos\alpha,\;\;\;\; [weber]=[tesla] [meter]^2$$
  
  
  This is the dot product of magnetic intensity vector $B$ and area vector $A$ (in the normal direction of the area), and $\alpha$ is the angle between the two directions.

  When $B$ and $A$ are in the same direction ($\alpha=0$), and if $B$ is 1 tesla and $A$ is 1 square meter, the flux $\phi$ is 1 weber.

• Electric Magnetic Interaction:

  In a magnetic field $\vec{B}$, a force $\vec{f}$ is exerted on a charge $q$ moving with velocity $\vec{u}$:
  

  $$\vec{f}=q\vec{u} \times \vec{B},\;\;\;\;\; [newton]=[coulomb]\frac{[meter]}{[second]} [tesla]$$
  
  
  where the force vector $\vec{f}$ is the cross product of velocity vector $\vec{u}$ and magnetic flux vector $\vec{B}$ (right-hand rule).

  A force of 1 newton is experienced by a charge of 1 coulomb moving with a velocity of 1 meter per second normal to a magnetic flux density of 1 tesla.