Ohm's law and its limitations, Current, Current density, Practice problems, FAQs

You turn on the switch of your fan, it starts spinning. You start the motor, the motor starts running almost instantaneously. So what do you think is responsible for making them work? The answer is electricity, also known as current. We cannot imagine a world without it. With that being said, let’s try to visualise current:a stream of positively charged particles moving from one point to another. Materials that allow current to flow through them are called conductors; you might have heard a lot about copper quoted as a good conductor, since it allows current to pass through it. But it's not as if the charges are going to flow in a conductor by themselves; there must be a potential difference between two points which causes current to flow; this potential difference is provided with the help of a battery. The correlation between current and potential difference was studied by a scientist named George Simon Ohm, who coined Ohm's law. In this article, we will explore current density, Ohm’s law and its limitations.

Table of contents

• Definition of current and current density
• Ohm’s law
• Limitations of Ohm’s law
• Practice problems
• FAQs

Definition of current and current density

• The amount of charge flowing divided by unit time is defined as current.
• Consider a conductor in which a total charge flows across a given cross section in a time Then the average value of current .
  
  
• The current flowing at a certain instant of time is called instantaneous current . Mathematically, it can be expressed as 

, where is the amount of charge flown in time

• The unit of charge is Coulomb and that of time is second ( Hence current carries a unit of or Ampere().
• If indicates charge of an electron,  

Note: 

• Direction of conventional current is from terminal of the battery to the negative; whereas the flow of electrons is from to terminal of the battery.
• Current is a scalar quantity.
  
  
• Current density( is defined as the current per unit area held normal to the surface. Let be the current flowing in an infinitesimally small amount of area . Then average current density can be expressed as, 

• Current density is a vector quantity.

• The instantaneous current density ; 

indicates the vector dot product of current density and area vector. 

indicates the area vector normal to the surface 

Ohm’s Law

• Consider a conductor which has length and area A battery having potential difference is connected across its ends and is the current flowing through it. 
• Ohm’s law states that the voltage supplied is proportional to the current flowing provided all physical conditions remain constant.

I.e

is called Resistance of the conductor. The unit of resistance is Ohm

• The electric field established in the conductor is given by,

The current density,

Now where indicates the conductivity of the material. 

(

.

indicates Ohm’s law

Comparing, we get

specific resistance or resistance of the conductor. From the above equation, it is clear that the resistance of a conductor is directly proportional to its length but inversely proportional to its area of cross section. The unit of resistivity is  

• Ohmic conductors (eg. Nichrome) are those which follow Ohm’s law. When temperature and other physical conditions are maintained constant, the graph between and for ohmic conductors is a straight line as shown:

Since , the slope of graph gives the value of reciprocal of resistance .

• Non-Ohmic conductors (eg. semiconductors, alloys) are those which do not obey Ohm’s law. The graph for such conductors is as shown: 
  
  
• As temperature increases, resistance of an ohmic conductor increases. The ions vibrate with more kinetic energy and the time between collisions reduces; this leads to an increase in resistance.

Limitations of Ohm’s law:

• Ohm’s law is not applicable to certain elements like diodes and transistors which allow current to flow in one direction only. 
• For certain electrical components like capacitors( used for storing charges) the variation of current and voltage is non-linear. Hence, Ohm’s law fails in such cases. 

Practice problems

Q. The current flowing through a conductor as a function of time is given by The charge that has flown in the wire between and is 

(a) (b) (c) (d)

A. c

Given, current

Now

Q. A potential difference of is applied between the ends of a conductor having a resistance of The charge flown through the wire in a time interval of min is 

(a) (b) (c) (d) 

A. b

Given

Q. In a conductor, it is found that electrons, each having a charge of pass from one point to another in Calculate the current flowing through the conductor? Also give the direction of the current? 

(a) (b) (c) (d)

A. a

Given, number of electrons,

Current,

The direction of conventional current is from terminal of the battery to terminal.

Q. A silver wire carries a current of Find the amount of charge that has flown in the wire in

A. Given, Current

Charge,

FAQs

Q. What is the value of ?
A. 1 ohm is the value of resistance of a conductor, when a voltage of applied causes a current of to flow through it.

Q. Why do we use current density?
A. It is useful for the design and construction of electrical and electronic elements. The strength of the current flowing through a given area helps to know the safe current level of the instruments. 

Q. How does increasing the radius of a wire affect the current density ?
A. 

Upon increasing the radius, the current density reduces. 

Q. Current is a scalar quantity. Why? 
A.  Current is scalar, because it follows algebraic addition or subtraction, but not vector addition or subtraction.