Q10P - Introduction to Electrodynamics

Question

(a) How do the components of a vectoii transform under a translation of coordinates (X= x, y = y- a, z = z, Fig. 1.16a)?

(b) How do the components of a vector transform under an inversion of coordinates (X= -x, y = -y, z = -z, Fig. 1.16b)?

(c) How do the components of a cross product (Eq. 1.13) transform under inversion? [The cross-product of two vectors is properly called a pseudovector because of this "anomalous" behavior.] Is the cross product of two pseudovectors a vector, or a pseudovector? Name two pseudovector quantities in classical mechanics.

(d) How does the scalar triple product of three vectors transform under inversions? (Such an object is called a pseudoscalar.)

Step-by-Step Solution

Answer

1. No component of the vector  $A$ is not changed during translation.
2. During inversion , $\overline{A}=-\overline{A}$ .
3. The cross product of two pseudo vectors is again a pseudo vector. The two pseudo vectors in mechanics are Torque and angular momentum.
4. Under inversion the scalar triple product of the pseudo vectors changes sign and , is inverted.

Step 1: Explain the concept of translation

The new position of the axes, after clockwise rotation are as $\overline{x}=x,\overline{y}=y-a$  and $\overline{z}=z$ . Let the vector $A$ be defined $A=A_{x}i+A_y j+A_{z}k$ . In the process of translation, each and every point of the object moves equally in same dimension along a straight line. Thus no component of the vector $A$ is not changed during translation.

Step 2: Describe the inversion of a vector

If the direction of a vector is $180°$ rotated wheras its magnitude remains same then, the inversion of vector happens. Thus inversion coordinates becomes $\overline{x}=-x,\overline{y}=-y$ and $\overline{z}=-z$ .

 

Therefore the new components of the vector $A$ are obtained as follows:

 

$$\begin{gathered} \overline{A_x}=-A_x \\ \overline{A_y}=-A_y \\ \overline{A_z}=-A_z \end{gathered}$$

 

It can be concluded that in the phenomena of inversion , $\overline{A}=-\overline{A}$ .

Step 3: Describe the inversion of a cross product

Let the two vector  $A$ and  $B$ are used in a cross product under inversion, as:

$\overline{A}\times \overline{B}$ . Under inversion, $\overline{A}=-\overline{A}$  and  $\overline{B}=-\overline{B}$.

 

Thus,  product $\overline{A}\times \overline{B}$ can be simplified as :

 

$$\begin{gathered} \overline{A}\times \overline{B}=-\overline{A}\times -\overline{B} \\ =\overline{A}\times \overline{B} \end{gathered}$$ 

 

Therefore there is no change in the cross product of two vectors under inversion. 

 

Let vectors  $A$ and  $B$ are pseudo vectors , that is , 

 

$$\begin{gathered} \left(\overline{A}\times \overline{B}\right)=\left(\overline{A} \right)\times \left(\overline{B} \right) \\ =\left(\overline{A}\times \overline{B}\right) \end{gathered}$$ 

Thus cross product of two pseudo vectors is again a pseudo vector. The two pseudo vectors in mechanics are Torque and angular momentum.

Step 4: Describe the inversion of a scalar triple product

Let the two vector  $A , B$ and  $C$ are pseudo vectors. Their scalar triple product is defined as $\overline{A}·\left(\overline{B}\times \overline{C} \right)$. Under inversion, $\overline{A}=-\overline{A}, \overline{B}=-\overline{B}$ and $\overline{C}=-\overline{C}$ .

Simplify $\overline{A}·\left(\overline{B}\times \overline{C} \right)$ under inversion.

 

$$\begin{gathered} \overline{A}·\left(\overline{B}\times \overline{C} \right)=-\overline{A}·\left(-\overline{B}\times -\overline{C} \right) \\ =\overline{A}·\left(\overline{B}\times \overline{C} \right) \end{gathered}$$ 

 

Thus under inversion the scalar triple product of the pseudo vectors changes sign and , is inverted.