Question

Three identical conducting balls A, B and C, each of mass \(m\), are thrown upward at an angle \(\theta\) to the horizontal with an initial speed \(v\) in a region of space that has a uniform electric field \(E\) downward along with the gravitational field \(g\). A is positively charged, B is uncharged and C is negatively charged. Rank the ranges \(R\) of these three balls in increasing order.

• A. \(R_A<R_B<R_C\)
• B. \(R_B<R_C<R_A\)
• C. \(R_A = R_B<R_C\)
• D. \(R_C<R_B<R_A\)

Hint:

For projectile motion, range is inversely proportional to effective downward acceleration. Greater downward acceleration gives smaller range.

Answer 1

The Correct Option is A

Step 1: Identify the forces.
All balls experience gravitational force downward. Since electric field is also downward, charged balls additionally experience electric force.

Step 2: Force on positively charged ball A.
For positively charged ball A, electric force acts in the direction of electric field, i.e. downward. So effective downward acceleration is greater than \(g\).
\[ g_A>g \]

Step 3: Force on uncharged ball B.
Ball B is uncharged, so it experiences only gravitational force. Its acceleration is:
\[ g_B = g \]

Step 4: Force on negatively charged ball C.
For negatively charged ball C, electric force acts opposite to the electric field, i.e. upward. So effective downward acceleration is less than \(g\).
\[ g_C<g \]

Step 5: Relation between range and effective acceleration.
For same initial speed and projection angle, projectile range is:
\[ R = \frac{v^2 \sin 2\theta}{g_{\text{eff}}} \]
Thus, range is inversely proportional to effective downward acceleration.

Step 6: Compare the ranges.
Since:
\[ g_A>g_B>g_C \]
Therefore:
\[ R_A<R_B<R_C \]
\[ \boxed{R_A<R_B<R_C} \] Hence, correct answer is option (A).