Ch 26: Potential and Field

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

Knight Calc 5th Edition

Ch 26: Potential and Field

문제 45

26장, 문제 45

The electric potential in a region of space is V=(150x² − 200y²)V, where x and y are in meters. What are the strength and direction of the electric field at (x, y)=(2.0 m, 2.0 m)? Give the direction as an angle cw or ccw (specify which) from the positive x-axis.

검증된 단계별 안내

The electric field **E** is related to the electric potential **V** by the negative gradient of the potential. Mathematically, this is expressed as:

E = -\(\nabla\) V

. The gradient operator in two dimensions is:

\(\nabla\) V = \(\left\)(\(\frac{\partial V}{\partial x}\), \(\frac{\partial V}{\partial y}\)\(\right\))

To find the x-component of the electric field, calculate the partial derivative of V with respect to x:

\(\frac{\partial V}{\partial x}\) = \(\frac{\partial}{\partial x}\)(150x^2 - 200y^2) = 300x

. Substituting x = 2.0 m, we get the x-component of the electric field.

To find the y-component of the electric field, calculate the partial derivative of V with respect to y:

\(\frac{\partial V}{\partial y}\) = \(\frac{\partial}{\partial y}\)(150x^2 - 200y^2) = -400y

. Substituting y = 2.0 m, we get the y-component of the electric field.

The magnitude of the electric field is given by:

|E| = \(\sqrt{E_x^2 + E_y^2}\)

, where

E_x

and

E_y

are the x- and y-components of the electric field, respectively. Substitute the values of

E_x

and

E_y

to compute the magnitude.

The direction of the electric field is given by the angle

\(\theta\) = \(\arctan\[\left\)(\(\frac{E_y}{E_x}\]\right\))

. Determine the angle and specify whether it is clockwise (cw) or counterclockwise (ccw) from the positive x-axis based on the signs of

E_x

and

E_y

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

주요 개념

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

Electric Potential

Electric potential, denoted as V, is the amount of electric potential energy per unit charge at a point in an electric field. It is a scalar quantity measured in volts (V) and indicates how much work would be done to move a charge from a reference point to a specific point in the field without any acceleration.

Electric Field

The electric field (E) is a vector field that represents the force experienced by a unit positive charge placed in the field. It is defined as the negative gradient of the electric potential, mathematically expressed as E = -∇V. The direction of the electric field is from regions of higher potential to lower potential.

Gradient and Direction

The gradient of a scalar field, such as electric potential, indicates the direction and rate of change of that field. In the context of electric fields, the gradient is calculated using partial derivatives with respect to spatial coordinates. The angle of the electric field can be determined using trigonometric functions based on the components of the field in the x and y directions.