The electric field in a region of space is E_x=5000x V/m , where x is in meters. Find an expression for the potential V at position x. As a reference, let V=0 V at the origin.

검증된 단계별 안내

Step 1: Recall the relationship between electric field and electric potential. The electric field is the negative gradient of the electric potential: \( E_x = -\frac{dV}{dx} \).

Step 2: Substitute the given electric field \( E_x = 5000x \) into the equation \( E_x = -\frac{dV}{dx} \). This gives \( -\frac{dV}{dx} = 5000x \).

Step 3: Rearrange the equation to solve for \( dV \): \( dV = -5000x \, dx \).

Step 4: Integrate both sides to find the potential \( V \). The integral of \( dV \) is \( V \), and the integral of \( -5000x \, dx \) is \( -2500x^2 \) (using the power rule for integration).

Step 5: Apply the boundary condition \( V = 0 \) at \( x = 0 \) to determine the constant of integration. Since \( V = -2500x^2 + C \), and \( V = 0 \) when \( x = 0 \), \( C = 0 \). Thus, the expression for the potential is \( V(x) = -2500x^2 \).

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주요 개념

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

Electric Field

The electric field (E) is a vector field that represents the force per unit charge experienced by a positive test charge placed in the field. It is defined mathematically as the negative gradient of the electric potential (V), indicating how the potential changes with position. In this case, the electric field is given as Ex = 5000x V/m, which shows that the field strength varies linearly with the position x.

Electric Potential

Electric potential (V) is a scalar quantity that represents the potential energy per unit charge at a point in an electric field. It is related to the electric field by the equation V = -∫E·dx, where the integral is taken along a path from a reference point to the point of interest. In this problem, we need to find the expression for V at position x, using the given electric field and the reference point where V = 0 V at the origin.

Integration in Physics

Integration is a fundamental mathematical tool used in physics to find quantities that accumulate over a continuous range, such as area under a curve or total work done. In the context of electric fields and potentials, integration allows us to calculate the potential difference by summing the contributions of the electric field over a distance. For this problem, we will integrate the electric field expression to derive the potential function V(x).