Electromagnetic Field Theory

Vector Calculus

ONE DIMENSIONAL PROBLEM = SCALAR

🌾Electric current has direction. But it is termed as a scalar. Why???

🌾If someone asks your weight, you will answer them as $70\; kg$ in scalar. But in physics book, weight is termed as a vector quantity. Why???

To answer the above questions, we need to classify the quantities based on their possible directions,

  • Quantities with NO direction
  • One-dimensional problem
    • Quantities with only ONE possible direction
    • Quantities with TWO possible (opposite) directions
  • Two-dimensional problem
  • Three-dimensional problem

🌾Quantities with NO direction:

For the quantities with no direction, we don’t have to define the direction. Hence quantities with no direction can be termed as SCALAR.

One dimensional problem:

🌾Quantities with only ONE possible direction:

$+x$ axis only

Example $1$

Imagine you and your friends are playing outside. Suddenly your friend says, 

” Hey dude… Look at the sky… It’s very dark.. I think it’s going to rain…” 

In this statement, he never mentioned about the direction of the sky (where to look).  He just said  “Look at the sky” not like “Look up at the sky”  Why?? 

Because, when someone said sky, by default we always look ‘up’.  Hence, there is no need to mention the direction.

‘Look at the sky’ is enough, no need for ‘Look up at the sky’… 

In the above example, the only possible direction is UPWARD.

Example $2$

Have you ever seen a milestone??

Why there is no direction sign in a milestone??

Because, wherever we see a milestone, there is only one way ahead of us. So, $300\;km$ in a milestone means that we have to travel in FORWARD direction for $300\;km$ to reach that place.

In this example, in milestone point of view, the only possible direction is FORWARD.

From the above two examples, we can say, if there is only one possible direction in a problem, we don’t need to define the direction.

Hence we can say, Quantities with only one possible direction = SCALAR.

🌾Quantities with only TWO possible directions:

Up – Down

Forward – Backward

$+x$ and $-x$ axes only (x-axis)

Example $1$

Imagine a milestone that says, 

Chennai $+100$km 

Madurai $-300$km

If you see this kind of milestone. What will be your guess???

  •  If we go in the same direction for $100km$, we will reach Chennai.
  •  If we go in the opposite direction for $300km$, we will reach Madurai.
 
From this we can say, Forward direction can be represented using positive number and Backward direction can be represented using negative number.

In the above example, the only TWO possible opposite directions are FORWARD and BACKWARD.

Example $2$

LIFT / ELEVATOR

The two possible (opposite) directions of an elevator are $$Up – Down$$ 

 See the lift buttons in the above image,

$0$ represents the ground floor,

$1$ represents the first floor,

$2$ represents the second floor and so on.

What about $-1$??

$-1$ and $-2$ represents the basement or underground floors. Right??

 

As ground floor as reference, upward floors are numbered in positive and downward basement is numbered in negative.

In the above example, the only TWO possible opposite directions are 

UPWARD and DOWNWARD

From the above two examples, we can say, if there are only two possible (opposite) directions involve in a problem, One direction can be represented using a positive number and the other opposite direction can be represented using a negative number.

The sign in the magnitude itself defines the direction.

Therefore no need to define the direction separately. 

Hence we can say, Quantity with only two possible (opposite) directions = SCALAR.

  • Quantities with NO direction – no need to mention the direction – hence SCALAR.
 
  • Quantities with only ONE possible direction ($+x$-axis only) – positive sign in the magnitude itself defines the direction – hence SCALAR.
 
  • Quantities with only TWO possible opposite directions (both $+x$ and $-x$ axes (x-axis only)) – positive sign in the magnitude defines one direction and negative sign defines the other direction – hence SCALAR.

Rephrasing the definition of scalar:

Some textbook definition of scalar is, 

Scalar is a physical quantity which has only magnitude and no direction.

But from above examples, quantities with only one possible direction and quantities with two possible opposite directions are also termed as scalar.

Even though these quantities have direction, there is no need to define it. In magnitude itself we are defining the direction.

So, the above textbook definition of scalar has to be rephrased as,

Scalar is a physical quantity that can be defined with its magnitude only