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Ch 26: Potential and Field
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 26: Potential and FieldProblema 45
Capitolo 26, Problema 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.

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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.

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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.
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Electric Potential

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.
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Intro to Electric Fields

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.
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Direction of Current in Loop Equations
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