Question

plz solve que no 18 and 19​

Answer 1

✪ᴀɴsᴡᴇʀ✪

1. MINIMUM VELOCITY

Given

• $\vec{E} = E\hat{i}$
• $\vec{B} = B\hat{j}$
• $\vec{F} = q\vec{E} + q\vec{v}\times \vec{B}$
• $\vec{F} = 0$

Velocity vector

Let $\vec{v}=v_x\hat{i}+v_y\hat{j}+v_z\hat{k}$

Then,

$\:\:\:\: \vec{F} = q\vec{E} + q\vec{v}\times \vec{B}$

$0= qE\hat{i} + q(v_x\hat{i}+v_y\hat{j}+v_z\hat{k})\times B\hat{j}$

$0= qE\hat{i} + q(v_xB\hat{k}-v_zB\hat{i})$

$0= q(E -v_zB)\hat{i} + qv_xB\hat{k}$

$v_z = \frac{E}{B}, v_x = 0$

$\vec{v}=v_y\hat{j}+\left(\frac{E}{B}\right)\hat{k}$

Magnitude of Velocity

Magnitude, v of the velocity is given by

$\to v =|v_y\hat{j}+\left(\frac{E}{B}\right)\hat{k}|$

$\to v =\sqrt{{v_y}^2+{\left(\frac{E}{B}\right)}^2}$

$\to v ≥ \sqrt{{\left(\frac{E}{B}\right)}^2}$

$\to v ≥ \frac{E}{B}$

• Thus particle should be sent with a minimum velocity of $\frac{E}{B}$.

Since final expression of $\vec{v}$ does not contain $\hat{j}$, they are perpendicular i.e.

• The particle should be sent perpendicular to the magnetic field direction.

2. An Example

Let

• $\vec{A} = 2\hat{i} + 3\hat{j}$
• $\vec{B} = 3\hat{i}$
• $\vec{C} = 2\hat{i} - \hat{j}$

It is clear that $\vec{A} ≠ \vec{C}$

Now

$\to \vec{A}\times \vec{B}$

$= (2\hat{i} + 3\hat{j})\times 3\hat{i}$

$= 6\hat{i}\times 3\hat{i} + 9\hat{j}\times \hat{i}$

$= 6 + 0$

$= 6$

$\to \vec{C}\times \vec{B}$

$= (2\hat{i} - \hat{j})\times 3\hat{i}$

$= 6\hat{i}\times 3\hat{i} - 3\hat{j}\times \hat{i}$

$= 6 + 0$

$= 6$

Thus

$\vec{A}\times \vec{B}= \vec{C}\times \vec{B}$