A small 1.00-g plastic ball that has a charge q of 1.00 C is...

A small 1.00-g plastic ball that has a charge $q$ of $1.00$ C is suspended by a string that has a length $L$ of $1.00 \mathrm{~m}$ in a uniform electric field, as shown in Figure 16-30. If the ball is in equilibrium when the string makes a $9.80^{\circ}$ angle with the vertical as indicated by $\theta$, what is the electric field strength? Example 16-5

Key Concepts

Force Equilibrium

In any system in equilibrium, the net force acting on the object is zero. This means that all forces, such as gravitational, tension, and electric forces, must balance out when summed vectorially. This balance is essential in setting up the equations that allow for the determination of unknown quantities, such as the electric field strength acting on a charged object.

Vector Decomposition

Forces acting at an angle are best analyzed by breaking them down into perpendicular components, usually horizontal and vertical. This decomposition allows one to apply equilibrium conditions independently along each axis, facilitating the calculation of individual force contributions and the resolution of the problem.

Electric Field and Charge Relationship

A charged object in an electric field experiences a force given by the equation F = qE, where q is the charge and E is the electric field strength. This relationship is fundamental in electrostatics and is used to relate the electric force to other forces acting on the object in equilibrium.

Trigonometric Relationships in Force Analysis

Trigonometric functions, particularly tangent, sine, and cosine, are used to relate the angles within a force diagram to the ratios of the force components. In equilibrium problems, these relationships help link the measured angle to the ratio of the horizontal (electric) force to the vertical (gravitational) force, leading to the determination of unknown variables like the electric field strength.

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