# Wright State University Lake Campus/2017-9/Phy2410/Brief equations T4

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#### 00-Mathematics for this course

Measured in radians, ${\displaystyle \theta =s/r}$ defines angle (in radians). The circumference of a circle is ${\displaystyle C_{\odot }=\,2\pi r}$ and the circle's area is ${\displaystyle A_{\odot }=\,\pi r^{2}}$ is its area. The surface area of a sphere is ${\displaystyle A_{\bigcirc }=4\pi r^{2}}$ and sphere's volume is ${\displaystyle V_{\bigcirc }={\frac {4}{3}}\pi r^{3}}$

#### 01-Introduction

• Earth's gravitational acceleration = g ≈ 9.8m/s2. Speed of light = c ≈ 3×108m/s. The electron has charge, e ≈ 1.6 × 10−19C and mass ≈ 9.11 × 10-31kg. 1eV = 1.602 × 10-19J is a unit of energy, defined as the work associated with moving one electron through a potential difference of one volt.
• ${\displaystyle k_{\mathrm {e} }={\frac {1}{4\pi \varepsilon _{0}}}\,}$≈ 8.987× 109 N·m²·C−2 is a fundamental constant of electricity; also ${\displaystyle \varepsilon _{0}={\frac {1}{4\pi k_{\mathrm {e} }}}\,}$ ≈ 8.854 × 10−12 F·m−1 is the vacuum permittivity or the electric constant.
• ${\displaystyle \mu _{0}\,}$ = 4π × 10−7 NA ≈ 1.257 × 10−6 N A (magnetic permeability) is the fundamental constant of magnetism: ${\displaystyle {\sqrt {\varepsilon _{0}\mu _{0}}}=1/c}$.

#### 18-Electric charge and field

• ${\displaystyle F=k_{\mathrm {e} }{\frac {qQ}{r^{2}}}={\frac {1}{4\pi \epsilon _{0}}}{\frac {qQ}{r^{2}}}\,}$ is Coulomb's law for the force between two charged particles separated by a distance r: ke≈8.987×109N·m²·C−2, and ε0≈8.854×10−12 F·m−1.
• ${\displaystyle {\vec {F}}=q{\vec {E}}}$ is the electric force on a "test charge", q.

Consider a collection of ${\displaystyle N}$ particles of charge ${\displaystyle Q_{i}}$, located at points ${\displaystyle {\vec {r}}_{i}}$ (called source points), the electric field at ${\displaystyle {\vec {r}}}$ (called the field point) is:

• ${\displaystyle {\vec {E}}({\vec {r}})={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}{\frac {{\widehat {\mathcal {R}}}_{i}Q_{i}}{|{\mathcal {\vec {R}}}_{i}|^{2}}}={\frac {1}{4\pi \varepsilon _{0}}}\sum _{i=1}^{N}{\frac {{\vec {\mathcal {R}}}_{i}Q_{i}}{|{\mathcal {\vec {R}}}_{i}|^{3}}}}$ is the electric field at the field point, ${\displaystyle {\vec {r}}}$, due to point charges at the source points,${\displaystyle {\vec {r}}_{i}}$ , and ${\displaystyle {\vec {\mathcal {R}}}_{i}={\vec {r}}-{\vec {r}}_{i},}$ points from source points to the field point.

${\displaystyle {\vec {E}}({\vec {r}})=k_{e}\int {\frac {{\hat {\mathcal {R}}}dQ}{{\mathcal {R}}^{2}}}}$ is the electric field due to distributed charge, where ${\displaystyle dQ\rightarrow \lambda d\ell \rightarrow \sigma dA\rightarrow \rho dV\;}$, and ${\displaystyle (\lambda ,\sigma ,\rho )}$ denote linear, surface, and volume density (or charge density), respectively. Line element:${\displaystyle d{\vec {\ell }}={\hat {x}}dx+{\hat {y}}dy+{\hat {z}}dz}$.

#### 19-Electric Potential and Electric Field

• ${\displaystyle U=qV}$ is the potential energy of a particle of charge, q, in the presence of an electric potential V.
• ${\displaystyle \Delta V=-E\ell \cos \theta ={\vec {E}}\cdot {\vec {\ell }}}$ (measured in Volts) is the variation in electric potential as one moves through an electric field ${\displaystyle E}$. If the field is not univorm, then ${\displaystyle \Delta V=-\sum {\vec {E}}\cdot \Delta \ell =-\int {\vec {E}}\cdot d\ell }$
• ${\displaystyle Q=CV}$ is the (equal and opposite) charge on the two terminals of a capacitor of capicitance, C, that has a voltage drop, V, across the two terminals.
• ${\displaystyle C=\varepsilon A/d}$ is the capacitance of a parallel plate capacitor with surface area, A, and plate separation, d. This formula is valid only in the limit that d2/A vanishes. If a dielectric is between the plates, then ε>ε0≈ 8.85 × 10−12 due to shielding of the applied electric field by dielectric polarization effects.
• ${\displaystyle U={\frac {1}{2}}QV={\frac {1}{2}}CV^{2}={\frac {Q^{2}}{2C}}}$ is the energy stored in a capacitor.
• ${\displaystyle \varepsilon _{0}\int {\vec {E}}\cdot {\vec {dA}}=Q_{encl}}$ is Gauss's law for the surface integral of the electric field over any closed surface, and ${\displaystyle Q_{encl}}$ is the total charge inside that surface.

#### 20-Electric Current, Resistance, and Ohm's Law

• ${\displaystyle I={\frac {\mathrm {d} Q}{\mathrm {d} t}}\ }$ defines the electric current. ${\displaystyle V=IR}$ is Ohm's Law relating I, and resistance, R, to the difference in voltage, V, between the terminals. ${\displaystyle P=IV=I^{2}R={\frac {V^{2}}{R}}}$ is the power dissipated as current flows through a resistor. Power is the rate at which energy is transformed.