Basic Quantities

• 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 (can be neither created nor destroyed).

• Force $f$: Force can be defined by Newton's second law:

  $$f=ma,\;\;\;\;\;\;\;[Newton]=\frac{[kilogram][meter]}{[second]^2}$$ (1)

  The force of 1 Newton will cause a mass of 1 kilogram to accelerate at 1 meter per second per second.
  • Gravitational force:

    $$f=G\frac{m_1m_2}{r^2}$$ (2)

    where $G=6.67384\times 10^{-11}$ $[Newton\,(meter/kilogram)^2]$ is the gravitational constant. In particular, on surface of Earth, the “weight” of a mass $m$ is

    $$f=G\frac{Mm}{r^2}=gm$$ (3)

    where $g=GM/r^2=9.8\;$meter$/$second$^2$ is the gravitational acceleration that measures the intensity of Earth's gravitational field, and $f$ is the force this field asserts on mass $m$.
  • Electric force:

    $$f=k\frac{q_1q_2}{r^2}=Eq_2$$ (4)

    where $k=8.98755\times 10^9$ $[Newton\,(meter/Coulumb)^2]$ is the Coulomb's constant, $E=kq_1/r^2$ is the intensity of the electric field caused by charge $q_1$, and $f$ is the force this field asserts on charge $q_2$.

• Potential Energy:
  • Gravitational Potential Energy: In the gravitational field of $g$, if a mass $m$ (with weight $f=mg$) is raised up from height $h_1$ to $h_2$, it receives a potential energy

    $$w=fl=mg(h_2-h_1),\;\;\;\;\;\;g=\frac{w}{m(h_2-h_1)},\;\;\;\; \frac{[Joule]}{[kilogram][meter]}=\frac{[Newton]}{[kilogram]} =\frac{[meter]}{[s]^2}$$ (5)

  • Electric Potential Energy: In an electric field of $E$, if a charge $q$ is moved along the direction of the field from point $l_1$ to point $l_2$, it receives a potential energy

    $$w=fl=qE(l_2-l_1),\;\;\;\;\;\;E=\frac{w}{q(l_2-l_1)},\;\;\;\; \frac{[Joule]}{[Coulomb][meter]}=\frac{[Newton]}{[Coulomb]}$$ (6)

• Current: The rate of flow of positive charge

  $$i=\frac{dq}{dt},\;\;\;\;\; q=\int i \; dt, \;\;\;\;\; [ ]=\frac{[Coulomb]}{[second]}$$ (7)

  which is one of the seven base units in the International System of Units (IS).

  The current is the same through out an electricity conducting component, it is a through variable.

  Current density $j=\lim_{A\rightarrow 0} i(A)/A$,

  $$q=\int i(t) \,dt =\int {\bf j} \cdot d{\bf A}\; dt$$ (8)

• Voltage: The potential energy per unit charge is measured by volt:

  $$v=v-v_0=\frac{w}{q}=\frac{Eq(l-l_0)}{q}=E(l-l_0), \;\;\;\;[Volt]=\frac{[Joule]}{[Coulomb]}$$ (9)

  where the voltege at $l_0$ (a reference point or “ground”) is $v_0=0$ by definition. The voltage is the difference between two potential energy levels $v_1$ and $v_2$ per unit charge in an electric field. A unit charge of $q=1$ Coulomb moved from one electric potential $v_1=El_1$ to another $v_2=El_2$ gains a potential energy

  $$v=v_2-v_1=E(l_2-l_1)$$ (10)

  The electric field can therefore also defined as:

  $$E=\frac{v}{l}=\frac{qv}{ql}=\frac{w}{ql}=\frac{fl}{ql}=\frac{f}{q}$$ (11)

  If a unit charge of $q=1$ Coulomb is moved through an electric field $E(l_2-l_2)$ of 1 volt, it receives or delivers 1 Joule of energy.

  Voltage $v=w/q$ is energy per charge, while electric field $E=f/q$ is force per charge.

  The voltage is measured as the difference across two points in an electric field or circuit (or a point with respect to a reference point called ground), i.e., it is an across variable.

• Power and Energy:

  Power $p$ is the rate of energy transformation. The transformation of 1 Joule of energy in 1 second represents a power of 1 Watt:

  $$p=\frac{dw}{dt}=\frac{dw}{dq} \; \frac{dq}{dt}=v\,i, \;\;\;\;\;\;... ...{[second]} =\frac{[Joule]}{[Coulomb]}\frac{[Coulomb]}{[second]} =[Volt][Ampere]$$ (12)

  We also have:

  $$w=\int p\;dt=\int vi\;dt, \;\;\;\;[Joule]=[Watt][second]=[Volt][Ampere][second]$$ (13)

  Another unit for power is horse power: $1 \;$   hp$=746 \;$   Watts$$

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

  • Electrical potential energy:

    The energy needed to move a charge $q$ from point A with potential $v_1$ to point B with potential $v_2$ is:

    $$w=qv=q(v_2-v_1)=qE(l_2-l_1)$$ (14)

    where the voltage $v=v_1-v_2=E(l_2-l_1)$ is the potential difference between the two points.

    An electric field $E=(v_2-v_1)/l$ is the electric potential difference per unit distance.

  • Gravitational potential energy:

    The energy needed to move a mass $m$ from height $h_1$ with potential $gh_1$ to height $h_2$ with potential $gh_2$ is

    $$w=fl=mg(h_2-h_1)$$ (15)

    where $gl=g(h_2-h_1)=gh_2-gh_1$ is the potential difference between the two heights. The Gravitational field $g=(gh_2-gh_1)/l$ is the gravitational potential difference per unit distance.

  

  • Electric force per unit charge is electric field $E=f/q$; Energy per unit charge is voltage $v=w/q=Eql/q=El$ is voltage.
  • Gravitational force per unit mass is gravitational field $g=f/m$; Energy per unit mass is $w/m=mgh/m=gh$. Although this quantity is not explicitly defined, it is analogous equivalent to voltage in electric field.

  Force	$f=GMm/r^2=gm$ $(g=GM/r^2)$	$f=kQq/r^2=Eq$ $(E=kQ/r^2)$
  Field intensity	$g=(gh_1-gh_2)/L=f/m$	$E=(v_2-v_1)/L=f/q$
  Potential difference	$gh_2-gh_1$	$v_2-v_1$
  Potential energy	$w=mg(h_2-h_1)=mgL$	$w=q(v_2-v_1)$

• Magnetic Flux:

  The intensity of magnetic effect (lines per unit area in a magnetic field or flux) is measured by magnetic flux density ${\bf B}$ in Tesla. The Earth's magnetic field is about 25 to 65 micro Tesla, the MRI machine is either 1.5 or 3 Tesla.

  The magnetic flux $\Phi$ through an area ${\bf A}$ (in the normal direction of the area) is

  $$\Phi=\int {\bf B} \cdot d{\bf A}=\vert B\vert\;\vert A\vert \cos\alpha,\;\;\;\; [Weber]=[Tesla] [meter]^2$$ (16)

  where $\alpha$ is the angle between the two directions.

  When ${\bf B}$ and ${\bf A}$ are in the same direction ($\alpha=0$), and if ${\bf B}$ is 1 Tesla and ${\bf A}$ is 1 square meter, then the flux $\Phi$ is 1 Weber.

• Electric Magnetic Interaction:

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

  $${\bf f}=q{\bf u} \times {\bf B},\;\;\;\;\; [Newton]=[Coulomb]\frac{[meter]}{[second]} [Tesla]$$ (17)

  where the force vector ${\bf f}$ is the cross product of velocity vector ${\bf u}$ and magnetic flux vector ${\bf B}$ (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.

  

  The Lorentz force on a charge $q$ in electromagnetic field is

  $${\bf f}=q({\bf E}+{\bf u}\times {\bf B})$$ (18)