Current Density-Definition, formula, Practice Problems, FAQs

You and your friends decide to conduct an experiment: you drain your laptops of charge, and they won’t boot. So you bring their adapters and plug it into the socket; you notice that the laptops start now. While you were doing this experiment, it is outside, so you’re sweating all over. You find an AC switch near to the socket where you put your laptop plug, you switch it on, and a cool breeze starts to fill the room thanks to the AC. So what is responsible for making the AC and the laptop function? The answer is electricity. Imagining electricity is easy; it is just the flow of charges from one place to another. A Greek scientist and mathematician, Thales of Miletus, around the BC, noticed that when amber rods were rubbed with animal fur, they produced a spark; this was proof of static electricity (electricity that does not flow). Later around scientist Benjamin Franklin experimented with a flying kite whose end was attached to an electrical wire. The wire acted as a “lightning rod”. He noticed that the rod becomes electrically charged when struck by lightning. This idea of electricity evolved over years thanks to experiments conducted by Faraday, Tesla, and so on. Current density gives an idea about the direction and number of charges in a current carrying conductor. In this article, we will explore current density in detail.

Table of contents

• Definition of current and current density
• Relation between current, drift velocity and conductivity
• Practice problems
• FAQs

Definition of current and current density

• The amount of charge flowing through a given cross section 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

•  Let number of charges per unit volume flow in a conductor. is the electronic charge. The conductor has an area of cross section and length Then the total charge in the conductor
• .

indicates the drift velocity of the electrons. It is the average velocity with which electrons get drifted when an external electric field () is applied.

• 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,  
  
  

• Direction of conventional current is from positive terminal to negative terminal in the circuit.
• Current density ( is defined as the amount of current flowing per unit area normal to the current flow. Let be the current flowing across an infinitesimally small amount of area . Then average current density can be expressed as, 

• Current density is a vector quantity.

Relation between current, drift velocity and conductivity

• ,

• is the relaxation time. It is the time between two consecutive collisions of electrons.
• ∴

sign indicates that the direction of the electric field and drift velocity are opposite.

Hence, 

 
is called the conductivity of the material. 

• In a circuit having a battery of emf volt and resistance the current flowing would be,

Note:

• Current is a scalar quantity.

Video explanation

Practice problems

Q1. The charge flowing through a conductor as a function of time is given by Calculate the current density of the wire at given that the area of cross section of the wire is ? 

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

Answer. c

Given, charge

Now

Q2. Which of the parameters has no effect on the drift velocity of electrons flowing through a conductor? 

(a) cross sectional area of the wire
(b) length of the wire
(c) The number of free electrons 
(d) magnitude of the current

Answer. b

Since current where is the drift velocity. 

It is independent of the length of the conductor.

Q3. A silver wire of diameter is connected to an aluminum wire of diameter A current of flows through the wires. Calculate the current density in the silver wire? Given that the number density of electrons in the wire is . 
Answer.

Radius of the silver wire,

Area of the wire,

Now current density

Q4. The current flowing through a wire is Calculate the current density of the wire through an area of cross section , whose normal makes an angle of to the current flow?
Answer.

Current density

FAQs

Question 1. Why do we use current density?
Answer. The strength of the current flowing through a given area helps to know the safe current level of the instruments. 

Question 2. Can current density be negative?
Answer. Yes, it can be negative. If the direction of flow of current is opposite to that of the area vector for the cross section chosen, then it is negative.

Question 3. Current is a scalar quantity. Why? 
Answer. Current is scalar, because it follows algebraic addition or subtraction, but not vector addition or subtraction. 

Question 4. What is the meaning of ?
Answer. 1 ohm is the value of resistance of a conductor, when a voltage of applied causes a current of to flow through it.