(a) The sign of the unknown charge is negative.
(b) The magnitude of the unknown charge is 2.18 μC.
What is charge?
The charge is the physical quantity which defines the object's electric field. The charged objects creates the electric field around it and attracts or repels another objects coming into that field.
Given is a charged ball of mass m = 0.265 kg and unknown charge q is hanging by a light thread from a ceiling. A fixed charge Q = +5.00 μC on an insulated stand is brought close to the unknown charge. As a result, the unknown charge hangs at an angle 0 = 38.0° to the vertical as shown in the diagram below. The distance between the two charges is r = 22.0 cm.
Tension in the string has two components, Tsinθ and Tcosθ
(a) The fixed charge Q will attract the movable charge q attached to string. The charge Q is positive. So, the charges must of opposite sign.
Thus, the sign of unknown charge is negative.
(b) From the equilibrium of forces, we get
Electrostatic force Fe = kQq/r²
Fe = Tsinθ .............(1)
and
weight force mg = Tcosθ.............(2)
Dividing both the equations, we get
mg/Fe = tanθ
mg / (kQq/r²) = tanθ
q = mgr²tanθ / kQ
Substitute the values from the question, we have
q = 0.265 x 9.81 x (0.22)² tan38 / (9x10⁹ x 5 x 10⁻⁶)
q = 2.18 x 10⁻⁶ C
or q = 2.18 μC
Thus, the magnitude of charge is 2.18 μC.
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