Current Electricity

SYNOPSIS

• Electric current: The rate of flow of electric charges through a particular cross section. The SI unit of current is Ampere.
• Ohm’s law: The current between the ends of a conductor is directly proportional to the potential difference applied across its end, provided all other factors remain constant. 
  V = IR
• Dependence of resistance on various factors:
  Resistance directly depends on length. Resistance is inversely proportional to area.
  With an increase in temperature, resistance increases.
• Current density: The amount of current flowing per unit area around that point, provided the area is normal to the direction of current.
• Drift speed, which is the magnitude of this velocity, is enormously small as compared to the thermal speed, which is not a vector and is much larger.
• Metals have low resistivity: Range of ρ varies from 10–8 Ω m to 10–6 Ω m.
  Insulators such as glass and rubber have high resistivity: Range of r varies from 1022 to 1024 times more than metals. 
  Semiconductors such as Si and Ge lie roughly in the middle range of resistivity on a logarithmic scale.
• When a conducting substance is brought under the influence of an electric field  free charges (e.g. free electrons in metals) move under the influence of this field in such a manner that the current density  due to their motion is proportional to the applied electric field.
  
  
  where σ is a constant of proportionality called electrical conductivity.
  
  
• Temperature coefficient of resistivity:
  For pure metals, resistance varies linearly with the rise of temperature.
  
  

• Emf (electromotive force) is the name given to a non-electrostatic agency. Typically, it is a battery in which a chemical process achieves this task of doing work in driving the positive charge from a low potential to a high potential.
  The effect of such a source is measured in terms of work done per unit charge in moving a charge once around the circuit. This is denoted by ε.

• Ohm’s law is obeyed by many substances, but it is not a fundamental law of nature. It fails if

• V depends on I non-linearly. Example: ρ increases with I even if temperature is kept fixed.
• The relation between V and I is non-unique. Example: GaAs
• The relation between V and I depends on the sign of V for the same absolute value of V.
• Kirchhoff's First Rule:
  At any junction of several circuit elements, the sum of currents entering the junction must equal the sum of currents leaving it.
  
  
  In the above junction, current I enters and currents I₁ and I₂ leave.
  Thus, I = I₁ + I₂.
  This is a consequence of charge conservation and assumption that currents are steady, i.e. no charge piles up at the junction.
• Kirchhoff's Second Rule:
  The algebraic sum of changes in potential around any closed resistor loop must be zero.  This is based on the principle that electrostatic forces alone cannot do any work in a closed loop, because this work equals the potential difference, which is zero, if we start at one point of the loop and come back to it.
  
  

  When applied to a loop as shown above (which could be part of a larger circuit), this gives
  -(R₁ + R₂) I₁ - R₃ I₃ - R₄ I₄ = 0.

• Points to remember in the case of current loops:
  Choose any closed loop in the network and designate a direction (in this example, counter clockwise) to traverse the loop.

• Go around the loop in the designated direction, adding emfs and potential differences. An emf is counted as positive when it is traversed from (−) to (+) and negative in the opposite case, i.e. from (+) to (−).

• An IR term is counted negative if the resistor is traversed in the same direction of the assumed current and positive if in the opposite direction.

• Equate the total sum to zero. 

• The Wheatstone bridge is an arrangement of four resistances—R₁, R₂, R₃ and R₄. The null point condition is given by 
  
  
  
  This is also known as the balance condition. If, for instance, R₁, R₂ and R₃ are known, then R₄ can be determined.
  

• In a balanced condition of the meter bridge,
  
  
  σ: Resistance per unit length of wire
  ℓ₁: Length of wire from one end where the null point is obtained
  
  A potentiometer is a device to compare potential differences. Because the method involves a condition of no current flow, the device can be used to measure potential differences and the internal resistance of a cell and to compare the emfs of two sources.