91Ó°ÊÓ

Electric fields are a fundamental concept in electromagnetism. They represent the force per unit charge exerted on a charged particle. Imagine them as invisible lines of force that influence charged objects in their vicinity. An electric field is described by vectors, which are characterized by both magnitude and direction.

• The direction of the field is the direction that a positive test charge would accelerate if placed within the field.
• The magnitude of the field is the force experienced per unit of charge.

Electric fields can be created by charged objects or can exist in empty space. Mathematically, an electric field \( \vec{E} \) can be expressed as the force \( \vec{F} \) divided by the charge \( Q \) as \( \vec{E} = \frac{\vec{F}}{Q} \). In our exercise, the electric field is given by \( \vec{E} = E_1 \hat{i} + E_2 \hat{j} \), which means it has components in the direction of the \( \hat{i} \) and \( \hat{j} \) unit vectors, typical of a two-dimensional plane.

In physics, 'work' refers to the energy transfer that occurs when a force moves an object over a distance. The work done by a force on an object can be calculated using the dot product of the force vector and the displacement vector. This is particularly relevant in the study of electric fields where the work done by an electric force on a charge can cause the charge to accelerate or move.

• The formula is: \{ W = \vec{F} \cdot \vec{r} \}.
• \( \vec{F} \) is the force vector while \( \vec{r} \) is the displacement vector.
• When dealing with electric fields, the force \( \vec{F} \) experienced by a charge \( Q \) is given by \( Q \vec{E} \).

Thus, the work done \( W \) in moving a charge in an electric field \( \vec{E} \) is \{ W = Q \vec{E} \cdot \vec{r} \} which simplifies to \( W = Q(E_1a + E_2b) \). This tells us how much energy is imparted to the charge as it moves within the field.

The dot product, also known as the scalar product, is a mathematical operation that takes two equal-length sequences of numbers—typically coordinate vectors—and returns a single number (a scalar). It is very useful in physics, particularly when dealing with vector quantities like force and displacement.

• For two vectors \( \vec{A} = A_1 \hat{i} + A_2 \hat{j} \) and \( \vec{B} = B_1 \hat{i} + B_2 \hat{j} \), the dot product is given by \( \vec{A} \cdot \vec{B} = A_1B_1 + A_2B_2 \).
• It results in a scalar, which simplifies calculations involving vectors.
• The dot product is also related to the cosine of the angle between two vectors, though in component form like our exercise, it is straightforward to calculate.

In the context of the exercise, we found \( \vec{E} \cdot \vec{r} = E_1a + E_2b \) by applying the definition of the dot product to the electric field vector and the displacement vector. This fundamental operation allows us to determine the work done by the electric force, simplifying to \( W = Q(E_1a + E_2b) \).