MEC O6

Analysis of Forces

The problem states the particle moves with constant speed (implying constant velocity to maintain zero net force) in the presence of $\vec{E}$, $\vec{B}$, and $\vec{g}$.

For the velocity to be constant, the net force must be zero:

$$ \vec{F}_{net} = q\vec{E} + q(\vec{v} \times \vec{B}) + m\vec{g} = 0 $$

Let’s define the coordinate system:

• $\vec{g} = -g\hat{k}$ (Vertical)
• $\vec{B} = B\hat{i}$ (Horizontal)
• $\vec{E} = E\hat{j}$ (Horizontal and perpendicular to $\vec{B}$)

Substituting these into the force equation:

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

Comparing coefficients:

• z-component ($\hat{k}$): $-mg – qv_y B = 0 \implies v_y = -\frac{mg}{qB}$
• y-component ($\hat{j}$): $qE + qv_z B = 0 \implies v_z = -\frac{E}{B}$

The x-component of velocity, $v_x$, is the component along the direction of the magnetic field. This is not determined by the force balance equation.

Analysis of Minimum Kinetic Energy

When the fields $\vec{E}$ and $\vec{B}$ are switched off, the particle moves under gravity alone (projectile motion) with initial velocity $\vec{v} = v_x\hat{i} + v_y\hat{j} + v_z\hat{k}$.

The minimum kinetic energy in projectile motion occurs at the highest point, where the vertical component of velocity ($v_z$) becomes zero. The speed at the highest point is determined by the horizontal components ($v_x$ and $v_y$).

Given: $K_{min} = \frac{1}{2} K_{initial}$

$$ \frac{1}{2} m (v_x^2 + v_y^2) = \frac{1}{2} \left[ \frac{1}{2} m (v_x^2 + v_y^2 + v_z^2) \right] $$ $$ 2(v_x^2 + v_y^2) = v_x^2 + v_y^2 + v_z^2 $$ $$ v_x^2 + v_y^2 = v_z^2 $$ $$ v_x^2 = v_z^2 – v_y^2 $$

Calculating $v_x$ (Component along $\vec{B}$)

Substitute the expressions for $v_y$ and $v_z$:

$$ v_x = \sqrt{ \left( -\frac{E}{B} \right)^2 – \left( -\frac{mg}{qB} \right)^2 } $$ $$ v_x = \sqrt{ \left( \frac{E}{B} \right)^2 – \left( \frac{mg}{qB} \right)^2 } $$

The direction can be parallel or anti-parallel to $\vec{B}$.