Two 4.0-g spheres are suspended by 9.0-cm-long light strings (see the figure). A uniform electric field is applied in the x-direction. If the spheres have charges of −3. 10-8 C and +3. 10-8 C, determine the electric field intensity that enables the spheres to be in equilibrium at θ = 8°. - ive tried everything please help
Is theta the angle between the spheres or the angle that each sphere string is tilted from vertical?
It makes a big difference.
To determine the electric field intensity required to maintain equilibrium, we can use Coulomb's Law and consider the forces acting on each sphere.
Let's start by calculating the gravitational force acting on each sphere. The weight of a sphere can be calculated using the formula:
Force_gravity = mass x gravity
Given that the mass of each sphere is 4.0 g, which is equivalent to 0.004 kg, and the acceleration due to gravity is approximately 9.8 m/s^2, we can calculate:
Force_gravity = 0.004 kg x 9.8 m/s^2
= 0.0392 N
Now, let's calculate the electrostatic force acting between the two spheres. Coulomb's Law states that the force between two charged objects is given by:
Force_electric = (k x |q1 x q2|) / r^2
Where k is the electrostatic constant (8.99 x 10^9 Nm^2/C^2), q1 and q2 are the charges of the spheres, and r is the distance between their centers. Given that q1 = -3.10^-8 C and q2 = 3.10^-8 C, and r = 9 cm = 0.09 m, we can calculate:
Force_electric = (8.99 x 10^9 Nm^2/C^2 x |(-3.10^-8 C) x (3.10^-8 C)|) / (0.09 m)^2
Note: The absolute value symbol is used because forces are always positive, regardless of the sign of the charges.
Calculating this expression gives us the value of the electrostatic force acting between the spheres.
Now, at equilibrium, the gravitational force and electrostatic force must cancel each other out. Since the spheres are in equilibrium at an angle of 8°, we need to find the component of the electrostatic force in the x-direction.
The x-component of the electrostatic force is given by:
Force_electric_x = Force_electric * cos(θ)
Where θ is the angle (in radians) between the electrostatic force and the x-axis. Since θ is given as 8°, we'll first convert it to radians:
θ_in_radians = (8°/ 180°) * π
Now, we can calculate the x-component of the electrostatic force:
Force_electric_x = Force_electric * cos(θ_in_radians)
Finally, to achieve equilibrium in the x-direction, the electric field intensity, E, must be equal to the x-component of the electrostatic force divided by the charge of one of the spheres:
E = Force_electric_x / q1
Now, assemble all the calculated values and solve for E to find the required electric field intensity.