Charges, circuits and optics

Start with two point charges

Rubbed materials can gain or lose electrons. An excess of electrons gives an object negative charge, and a deficit gives positive charge. For two stationary point charges in vacuum, the magnitude of the electrostatic force is $F=k|q_1q_2|/r^2$. Here $r$ is their separation. The sign of the charge product tells you which way to draw the forces. Like charges repel and unlike charges attract. The two forces have equal magnitudes even when the charges have different magnitudes. Each acts on a different object.

For a comparison, write the new force divided by the old force before inserting any numbers. Increasing one charge by a factor of three triples the numerator. Increasing the separation by a factor of three multiplies the denominator by nine. Doing both therefore reduces the force to one third of its initial value. The constant cancels because the medium stays the same. A quick direction check is to hold the charges fixed and picture them moving very far apart. The calculated force should tend towards zero.

The electric field at a point is the force per unit positive test charge there. A negative test charge feels a force opposite to the field. The field around a single positive charge points outwards. To combine fields from two charges, draw the arrows at the point where you want the field. Equal magnitudes there will cancel only if their directions are opposite.

Mark the junctions in a circuit

Current measures charge flowing per unit time, and voltage measures potential difference between two points. An ohmic resistor obeys $V=IR$ at a given temperature. Label the two ends of each resistor before reducing a network. Resistors are in series (in serie) when they carry the same current along an unbranched path. The voltage drops then add, which gives an equivalent resistance equal to the sum. A wire drawn with a bend still connects the same points, so the physical appearance of the diagram can be misleading.

Resistors in parallel (in parallelo) share the same two end nodes. Each branch experiences the full voltage between those nodes. Branch currents add to the current supplied, which gives $1/R_{\rm eq}=1/R_1+1/R_2$ for two branches. An extra parallel branch gives the charge a second conducting path, so the equivalent resistance is smaller than either positive branch resistance. Two identical resistors in parallel have half the resistance of either one alone. Calculating the branch currents and adding them often takes less arithmetic than adding fractions.

Electrical power is the rate at which energy transfers. For a resistor it is $P=VI$, using the voltage across and current through that same resistor. Substituting Ohm's law gives $I^2R$ or $V^2/R$. These describe different comparisons. At fixed current a larger resistance dissipates more power. At fixed voltage a larger resistance carries less current and dissipates less power. To find energy supplied during a steady interval, multiply power in watts by time in seconds to obtain joules.

Draw a normal at the boundary

When a light ray strikes a plane surface, draw the normal, a line perpendicular to that surface through the point of incidence. Reflection sends the ray back into the first medium at the same angle to the normal as the incident ray. If the stated angle is measured from the surface, subtract it from a right angle before using a formula. At perpendicular incidence the angle to the normal is zero, although the ray meets the surface at a right angle.

Refraction (rifrazione) changes the direction of an oblique ray transmitted between media with different refractive indices. Snell's law gives $n_1\sin\theta_1=n_2\sin\theta_2$. Entering the larger index makes the sine smaller, so the ray bends towards the normal. Leaving the larger index can bend it away. If the computed sine exceeds one, there is no transmitted ray in the second medium and total internal reflection occurs. For hand calculations you can often compare the sines or leave the requested result as a fraction without finding an angle numerically.