Formula sheet: Electric Charges and Fields

Electric Charges and Fields14 min readLesson 1 of 8

1.1 Forces and fields Two point charges at rest act on each other with a force that acts along the line joining them. Its size grows with the product of the charges and falls with the square of the distance between them. The constant that sets the strength of the force is written as k, and in SI units k is a little under 9 \times 10^9 N m^2 C^-2. The force on charge 1 due to charge 2 is a vector, so the direction has to be carried through every superposition problem. Keep the convention that the unit vector points from the source charge to the charge that feels the force. At any point in space a charge q sets up an electric field. Test the field by placing a small positive charge there and dividing the force it feels by its own charge. The field of a point charge therefore points away from a positive source and toward a negative source.

F = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r ^{2}}

Magnitude of the force between two point charges in vacuum

F_{12} = \frac{1}{4 π ε _{0}} \frac{q _{1} q _{2}}{r _{12}^{2}} r^_{12}

Coulomb force on charge 1 due to charge 2; the unit vector points from 2 to 1

ε_{0} = 8.854 \times 1 0^{- 12} C^{2} N^{- 1} m^{- 2}, \frac{1}{4 π ε _{0}} = 9 \times 1 0^{9} N m^{2} C^{- 2}

Permittivity of free space and the Coulomb constant

E = \frac{F}{q _{0}} = \frac{1}{4 π ε _{0}} \frac{q}{r ^{2}} r^

Electric field of a point charge, in N C^-1

F_{medium} = \frac{F _{vacuum}}{K}

A dielectric of dielectric constant K weakens the force by that factor

1.2 Dipole, flux and Gauss's law Picture a dipole as two equal charges of opposite sign sitting very close together. The moment of the pair runs from the negative end to the positive end, and the fields from the two charges nearly cancel at distant points, so the net field drops faster than the field of a lone charge. The torque on a dipole in a uniform field turns the dipole toward the field direction, and the potential energy is lowest when the moment is parallel to the field. The electric flux through a surface counts how much of the field passes through it. For a closed surface the flux fixes the net charge inside, whatever the shape of the surface. The three standard applications use a cylinder around a line charge, a pillbox across a sheet, and a sphere around a shell. Quantity Expression Where it applies Field of an infinite line charge k 2 lambda / r distance r from a long wire, lambda is charge per length Field of an infinite plane sheet sigma / (2 epsilon0) any point near a large sheet, sigma is charge per area Field outside a spherical shell k Q / r^2 r greater than the radius, behaves like a point charge Field inside a spherical shell zero r less than the radius Flux through a closed surface 1 / epsilon0 times net charge inside any closed surface at all 1.3 Units and constants you should not lose in the working Charge is measured in coulomb, the constant k in N m^2 C^-2, the field in N C^-1, flux in N m^2 C^-1, dipole moment in C m and torque in N m. Convert every length to metre before substituting, and keep the sign of the charge when a vector direction matters.

Quantity	Expression	Where it applies
Field of an infinite line charge	k 2 lambda / r	distance r from a long wire, lambda is charge per length
Field of an infinite plane sheet	sigma / (2 epsilon0)	any point near a large sheet, sigma is charge per area
Field outside a spherical shell	k Q / r^2	r greater than the radius, behaves like a point charge
Field inside a spherical shell	zero	r less than the radius
Flux through a closed surface	1 / epsilon0 times net charge inside	any closed surface at all

Exam

In a working draw the charges, mark the direction of the force or field on the diagram, and only then substitute numbers. Most lost marks in this chapter come from a cm left unconverted or from a field direction reversed.

in hinglish

Is page ka kaam hai formula aur condition ko jodna. Coulomb's law mein force q1 q2 ke product ke saath badhta hai aur r ke square ke saath ghatta hai, aur medium aane par force dielectric constant K se kam ho jata hai. Electric field define hota hai force per unit charge se, aur point charge ka field k q / r^2 hai. Dipole moment ki direction negative se positive charge ki taraf hoti hai. Axial point par field 2kp / r^3 aur equatorial point par kp / r^3 hota hai, isliye axial field equatorial field ka do guna hai. Torque ka formula p E sin theta hai aur potential energy -p E cos theta hai. Gauss's law mein closed surface ka flux sirf andar ke net charge par depend karta hai, surface ki shape par nahi. Wire ka field lambda / (2 pi epsilon nought r), sheet ka field sigma / (2 epsilon nought) aur shell ke bahar field point charge jaisa hota hai, shell ke andar zero. Numbers lagane se pehle sab lengths metre mein badal lo aur diagram par direction bana lo.