Question

A charge q=4muC has as instantaneous velociyt v=(2hati-3hatj+hatk)xx10^6m/s in a uniform magnetic field B=(2hati+5hatj-3hatk)xx10^-2T. What is the force on the charge?

Answer 1

Hence the value of force on the charge is − 0.16 i ^  − 0.32  j ^  + − 0.64  k ^

Explanation:

Given data:

• q = 4 m.u.C
• v = (2 hati - 3 hat j + hat k) x 10^6 m/s
• B = (2 hati + 5 hat j - 3 hat k) x 10^-2 T

F  = q  v  ×  B  

Where q has a positive or negative sign

On substituting, we get

− 0.16 i ^  − 0.32  j ^  + − 0.64  k ^

Hence the value of force on the charge is − 0.16 i ^  − 0.32  j ^  + − 0.64  k ^

Answer 2

Given :

The magnitude of charge = q = 4 $\mu$ c

The instantaneous velocity = v =  ( $\vec{2i}$ -  $\vec{3j}$ + $\vec{k}$ ) × $10^{6}$  m/s

The magnitude of magnetic field = B =  ( $\vec{2i}$ +  $\vec{5j}$ - $\vec{3k}$ ) × $10^{-2}$  T

To Find :

The force on the charge

Solution :

Let The force on the charge = F

∵ Force  = q  v  ×  B  

Where q is the charge with positive or negative sign

            v is the charge velocity

            B is the magnetic field

Now,

 Substitute the value of q v B

So,   F = q × v  ×  B  

          = 4 $\mu$ c  × (  $\vec{2i}$ -  $\vec{3j}$ + $\vec{k}$ ) × $10^{6}$  m/s  ×   ( $\vec{2i}$ +  $\vec{5j}$ - $\vec{3k}$ ) × $10^{-2}$  T

    Applying vector property

       i.e  ( i . i ) = ( j . j ) = ( k . k ) = 1

      And (i . j ) = ( j .k) = ( k . i ) = 0

∴    F = 4 $\mu$ c  × (  $\vec{2i}$ -  $\vec{3j}$ + $\vec{k}$ ) × $10^{6}$  m/s  ×   ( $\vec{2i}$ +  $\vec{5j}$ - $\vec{3k}$ ) × $10^{-2}$  T

        =  4 × [ ($\vec{2i}$ . $\vec{2i}$) + ($\vec{2i}$ . $\vec{5j}$) - ($\vec{2i}$ . $\vec{3k}$) - ( $\vec{3j}$ . $\vec{2i}$ ) - ( $\vec{3j}$ . $\vec{5j}$ ) + ($\vec{3j}$ . $\vec{3k}$) + ($\vec{k}$ . $\vec{2i}$ ) +

            ($\vec{k}$ . $\vec{5j}$ ) - ( $\vec{k}$ . $\vec{3k}$) ]

        = 4 × [ 4 + 0 - 0 - 0 - 15 + 0 + 0 + 0 - 3 ]

        = 4 × ( 4 - 15 - 3 )

        = 4 × ( 4 - 18 )

        = 4 × ( - 14 )

i.e Force = - 56

So, The Force on the charge = F = $\left | -56 \right |$  = 56 N

Hence, The Force on the charge is 56  N    Answer