Q44P - Fundamentals Of Physics

Question

Question: An alpha particle (the nucleus of a helium atom) has a mass of $\mathbf{6}\mathbf{.}\mathbf{64}\mathbf{\times }\mathbf{ }{\mathbf{10}}^{\mathbf{-}\mathbf{27}}\mathbf{kg}$ and a charge of +2e. What are the (a) magnitude and (b) direction of the electric field that will balance the gravitational force on the particle?

Step-by-Step Solution

Answer

• a) The magnitude of the electric field that will balance the gravitational force on the particle is $2.03\times {10}^{-7}N/C$
• b) The magnitude of the electric field that will balance the gravitational force on the particle is upward.

Step 1: The given data

Mass of the alpha particle,$m=6.64\times {10}^{-27}kg$

Charge of the alpha particle,$q = +2e$

Step 2: Understanding the concept of electric field and gravitational force

The force on the object due to gravity, also called as weight of the object, is equal to the product of mass and the gravitational acceleration. When the charged particle is placed in an electric field, the electrostatic force acting on the particle is equal to the product of charge and electric field.

 

Using the concept of force due to gravity on the alpha particle and relating the value to the electric field produced by the particle, we can get the magnitude of the electric field with its direction.

Formulae:

The force due to gravity, F = mg                     (i)

The force and electric field relation, F = qE   (ii)

Step 3: a) Calculation of the magnitude of the electric field

Vertical equilibrium of forces leads to the balance of the forces due to gravity and due to electric filed. Thus, using equation (i) in equation (ii), we can get the magnitude of the electric field as follows:

$$\begin{gathered} \left|\underset{E}{\to }\right|= \frac{mg}{2e} \\ =\frac{\left(6.64\times {10}^{-27}kg\right)\left(9.8 m/s^2\right)}{2(1.60\times {10}^{-19}C)} \\ =2.03\times {10}^{-7} N/C \end{gathered}$$

 

Hence, the value of the electric field is$2.03\times {10}^{-7}N/C$

Step 4: b) Calculation of the direction of the electric field

Since the force of gravity is downward, then the force due to the electric field must point upward. Since q > 0 in this situation, this implies the electric field must itself point upward.