Metal sphere \(A\) has a radius of \(5 \mathrm{~cm}\) and metal sphere \(B\) has a radius of \(10 \mathrm{~cm}\). Sphere \(A\) carries a charge of \(9 \mathrm{nC}\) and sphere \(B\) carries a charge of \(18 \mathrm{nC}\). If the surfaces of \(A\) and \(B\) are \(185 \mathrm{~cm}\) apart, the potential energy between them is (A) \(7.29 \times 10^{-17} \mathrm{~J}\) (B) \(7.88 \times 10^{-9} \mathrm{~J}\) (C) \(7.88 \times 10^{-7} \mathrm{~J}\) (D) \(7.29 \times 10^{-9} \mathrm{~J}\) (E) None of the above

Electric charge is a fundamental property of matter that explains how objects interact electromagnetically. There are two types of charge, positive and negative. Like charges repel each other while opposite charges attract.

Electric charges can be transferred between objects, but they cannot be created or destroyed. The amount of charge is measured in coulombs (C). Small charges are often measured in subunits like nanocoulombs (nC), where 1 nC equals to 1 billionth of a coulomb, or \(10^{-9}\) C. For example, if a sphere carries a charge of 9 nC, it has a charge of \(9 \times 10^{-9}\) C.

Understanding electric charge is crucial for grasping other concepts in physics, such as electric fields, electrical potential, and Coulomb's law. Additionally, being able to calculate the effects of charge interactions is vital for solving problems on standardized tests like the SAT or AP Physics exams.

Coulomb's law describes the electrostatic interaction between electrically charged particles. According to this law, the force \(F\) between two point charges is directly proportional to the product of their charges \(q_1\) and \(q_2\), and inversely proportional to the square of the distance \(r\) between them.

The formula is written as:
\[F = k \frac{q_1 \cdot q_2}{r^2}\]
where \(k\) is Coulomb's constant (\(8.99 \times 10^9 \mathrm{N\cdot m^2/C^2}\)).

Significance in Calculations

Coulomb's law is fundamental in calculating the force between charged particles. Additionally, it helps us determine the electric potential energy, which is energy due to the position of charged objects in a field. A solid grasp of Coulomb's law is also important for students preparing for standardized tests like the SAT Physics subject test or the AP Physics exam.

Electric potential energy (U) is the energy that charged objects possess by virtue of their position in an electric field. For two point charges, the electric potential energy is given by the formula:
\[U = k \frac{q_1 \cdot q_2}{d}\]
where \(k\) is the Coulomb's constant, \(q_1\) and \(q_2\) are the amounts of charge, and \(d\) is the separation between the charges.

This concept is analogous to gravitational potential energy, where energy is stored due to an object's position in a gravitational field. Understanding this formula is essential for solving problems related to electrostatics in physics and is often tested in SAT and AP physics exams.

Preparation for the SAT Physics subject test involves a thorough understanding of key concepts like electric charge, Coulomb's law, and electric potential energy. These topics are among the fundamentals of electromagnetism, which is a significant part of the exam.

Tips for SAT Physics:

• Review electric concepts methodically and understand their applications.
• Practice problems that involve calculations and conceptual understanding.
• Use flashcards to remember important formulas and constants.
• Take timed practice tests to become familiar with the exam format and improve speed.

Comprehensive preparation will help ensure that students are well-equipped to tackle questions related to the potential energy between charged spheres and other related topics.

AP Physics courses, such as AP Physics 1 and AP Physics C, cover a wide range of topics, including electricity and magnetism. In the AP review:

Key Focus Areas

• Review foundational electric concepts, as these are pivotal for problem-solving.
• Perform experiments to understand the practical aspects of electrostatics.
• Work through past AP exam questions to gain insight into the question types and difficulty level.
• Pay special attention to how electric potential energy is derived from Coulomb's law in the context of different charge configurations.

By focusing on these areas, students can hone their understanding of physics principles and excel in their AP exams, particularly when dealing with problems related to electric charges and their interactions.