Two positive point charges are a fixed distance apart. The sum of their charges is Qₜ. What charge must each have in order to
(a) maximize the electric force between them, and
(b) minimize it?

검증된 단계별 안내

Step 1: Recall Coulomb's Law, which states that the electric force between two point charges is given by \( F = \frac{k \cdot q_1 \cdot q_2}{r^2} \), where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them.

Step 2: Express the relationship between the charges \( q_1 \) and \( q_2 \) using the given total charge \( Q_t \). Since the sum of the charges is \( Q_t \), we can write \( q_1 + q_2 = Q_t \). Let \( q_1 = x \) and \( q_2 = Q_t - x \).

Step 3: Substitute \( q_1 = x \) and \( q_2 = Q_t - x \) into Coulomb's Law to express the force as a function of \( x \): \( F(x) = \frac{k \cdot x \cdot (Q_t - x)}{r^2} \). Simplify this to \( F(x) = \frac{k \cdot Q_t \cdot x - k \cdot x^2}{r^2} \).

Step 4: To maximize the force, find the value of \( x \) that maximizes \( F(x) \). This requires taking the derivative of \( F(x) \) with respect to \( x \), setting it equal to zero, and solving for \( x \). The derivative is \( \frac{dF}{dx} = \frac{k \cdot Q_t}{r^2} - \frac{2k \cdot x}{r^2} \). Set \( \frac{dF}{dx} = 0 \) and solve for \( x \).

Step 5: To minimize the force, consider the extreme cases where one charge approaches zero (\( q_1 \to 0 \)) and the other approaches \( Q_t \) (\( q_2 \to Q_t \)), or vice versa. Substitute these values into the force equation \( F = \frac{k \cdot q_1 \cdot q_2}{r^2} \) to confirm that the force approaches zero in these cases.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

12m

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Coulomb's Law

Coulomb's Law describes the electrostatic force between two point charges. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is fundamental in understanding how charges interact, as it quantifies the strength and direction of the force based on charge values and separation distance.

Charge Distribution

Charge distribution refers to how total charge Qₜ is allocated between two point charges. To maximize the electric force, both charges should be equal, meaning each charge would be Qₜ/2. Conversely, to minimize the force, one charge should be zero, leaving the other with the entire charge Qₜ, as the force depends on the product of the charges.

Electric Force Maximization and Minimization

Maximizing and minimizing electric force involves understanding how the configuration of charges affects the resultant force. The maximum force occurs when both charges are equal, maximizing their product. In contrast, the minimum force occurs when one charge is zero, resulting in no interaction, illustrating how charge values directly influence the strength of the electric force.