FIGURE P23.44 shows a thin rod of length L with total charge Q. Evaluate E at r=3.0 cm if L=5.0 cm and Q=3.0 nC.

검증된 단계별 안내

Understand the problem: We are tasked with finding the electric field (E) at a distance r = 3.0 cm from a thin rod of length L = 5.0 cm and total charge Q = 3.0 nC. The rod is uniformly charged, so the charge per unit length (λ) is constant. The electric field at a point due to a charged rod can be calculated using integration.

Step 1: Define the charge distribution. The linear charge density (λ) is given by λ = Q / L. Substitute the given values for Q and L to find λ.

Step 2: Set up the expression for the electric field. The electric field at a point due to a small charge element (dq) on the rod is given by dE = (1 / (4πε₀)) * (dq / r²), where dq = λ dx and r is the distance from the charge element to the point of interest. Here, ε₀ is the permittivity of free space.

Step 3: Integrate to find the total electric field. Since the rod is thin and lies along a straight line, the electric field components perpendicular to the rod cancel out, leaving only the horizontal component. The total electric field is E = (1 / (4πε₀)) * ∫(λ dx / r²), where the limits of integration are from 0 to L.

Step 4: Evaluate the integral. Substitute λ = Q / L and r = 3.0 cm into the integral. Perform the integration to find the electric field E. Finally, substitute the numerical values for Q, L, r, and ε₀ to calculate the magnitude of E.

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주요 개념

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

Electric Field (E)

The electric field (E) is a vector field that represents the force per unit charge experienced by a positive test charge placed in the vicinity of electric charges. It is defined mathematically as E = F/q, where F is the force acting on the charge and q is the magnitude of the charge. The direction of the electric field is away from positive charges and towards negative charges.

Linear Charge Density (λ)

Linear charge density (λ) is defined as the amount of charge per unit length along a charged object, such as a rod. It is calculated using the formula λ = Q/L, where Q is the total charge and L is the length of the rod. This concept is crucial for determining the electric field produced by a uniformly charged rod.

Coulomb's Law

Coulomb's Law describes the electrostatic interaction between charged objects. It states that the electric force (F) between two point charges is directly proportional to the product of the magnitudes of the charges (q1 and q2) and inversely proportional to the square of the distance (r) between them, expressed as F = k * (q1 * q2) / r², where k is Coulomb's constant. This law is fundamental in calculating the electric field generated by charge distributions.