State the formula giving relation between electric field intensity and potential gradient

F = 1 4 π ϵ 0 q Q r A 2 F = \dfrac F = 4 π ϵ 0 ​ 1 ​ r A ​ 2 q Q ​ F, equals, start fraction, 1, divided by, 4, pi, \epsilon, start subscript, 0, end subscript, end fraction, start fraction, q, Q, divided by, r, start subscript, A, end subscript, squared, end fraction Gravity is conservative. When you lift a book up, you do work on the book. If you gently lower the book back down, the book does work on you. The net amount of work is zero. You can raise and lower a hundred times, and if the book ends up in the original height, the net amount of work is zero. If you move the book horizontally, the amount of work is also zero, because there is no opposing force in the horizontal direction. For both gravity and electricity, potential energy differences are what's important. Wherever your book starts out, it has some potential energy. When you move the book, you add or remove potential energy relative to where it started. For moving charges, you add or subtract electric potential energy relative to where the charge started. If you wonder if an object is storing potential energy, take away whatever might be holding it in place. If the object moves, it was storing potential energy. An apple falls from a tree and conks you on the head. It had potential energy. Let go of a charge in an electric field; if it shoots away, it was storing electric potential energy. Let's work it out: The amount of work done is force times distance, W = F ⋅ d W = F \cdot d W = F ⋅ d W, equals, F, dot, d...

\( \newcommand\) has a useful physical interpretation. Recall that the gradient of a scalar field is a vector that points in the direction in which that field increases most quickly. Therefore: The electric field points in the direction in which the electric potential most rapidly decreases. This result should not come as a complete surprise; for example, the reader should already be aware that the electric field points away from regions of net positive charge and toward regions of net negative charge (Sections 2.2 and/or 5.1). What is new here is that both the magnitude and direction of the electric field may be determined given only the potential field, without having to consider the charge that is the physical source of the electrostatic field. Example \(\PageIndex \nonumber \] as was determined in Section 5.1.

In electrostatics, the concept of Electric field and electric potential plays an important role. Electric field or electric field intensity is the force experienced by a unit positive test charge and is denoted by E. Electric potential is the Mathematically, the electric field and the potential is given by: ⇒\[E=\frac=6x\] At x=2, ⇒ E=6(2)=12 V/m Therefore, the electric field E at (2,1,2) is 12V/m. Electricity Keeping us Alive Everyone is familiar with electricity in modern times. We get electricity from sockets in walls in our houses. It gives us a shock if we touch it. Science has taught us that all matter is made up of very small particles called atoms. These atoms are made of two kinds of charge – positive and negative. The middle part of the atom contains the positive charge and flying around this is the negative charge. Most of the time the number of positive and negative charges in an atom are equal in number – they exist in pairs. When they are not equal in number, the extra negative charge leaves the atom and goes looking for its partner. These stray charges are electrons and are easier to move about. These moving electrons make up electricity. Types of Electricity Electricity is of two kinds – static and current. Static electricity, electrons are moved mechanically. In current electricity, the electrons move in a closed loop. When the loop is broken, electricity cannot flow. Lightning is static electricity. During thunderstorms, a cloud can develop a buildup of n...

Let us consider two closely spaced equipotential surfaces A and B as shown in figure.Let the potential of A be V A ​ = V and potential of B be V B ​ = V − d V is decrease in potential in the direction of electric field E normal to A and B. Let dr be the perpendiculare distance between the two equipotential surfaces. when a unit positive charge is moved along this prependicular from the surface B to surface A against the electric field, the work done in this process is W B A ​ = − E ( d r )

https://phys.libretexts.org/@app/auth/3/login?returnto=https%3A%2F%2Fphys.libretexts.org%2FBookshelves%2FUniversity_Physics%2FBook%253A_University_Physics_(OpenStax)%2FBook%253A_University_Physics_II_-_Thermodynamics_Electricity_and_Magnetism_(OpenStax)%2F07%253A_Electric_Potential%2F7.05%253A_Determining_Field_from_Potential Learning Objectives By the end of this section, you will be able to: • Explain how to calculate the electric field in a system from the given potential • Calculate the electric field in a given direction from a given potential • Calculate the electric field throughout space from a given potential Recall that we were able, in certain systems, to calculate the potential by integrating over the electric field. As you may already suspect, this means that we may calculate the electric field by taking derivatives of the potential, although going from a scalar to a vector quantity introduces some interesting wrinkles. We frequently need \(\vec\) equals the rate of decrease of V with distance. The faster V decreases over distance, the greater the electric field. This gives us the following result. Relationship between Voltage and Uniform Electric Field In equation form, the relationship between voltage and uniform electric field is \[E = - \dfrac\] Example \(\PageIndex\): Electric field vectors inside and outside a uniformly charged sphere. Example \(\PageIndex\] Significance Again, this matches the equation for the electric field found previously. It also de...

Relation between Electric Field and Electric Potential The electric field exists if and only if there is an electric potential difference. If the charge is uniform at all points, however high the electric potential is, there will not be any electric field. Thus, the relation between electric field and electric potential can be generally expressed as – “Electric field is the negative space derivative of electric potential.” Electric Field and Electric Potential The relation between the electric field and Direction of Electric Field • If the field is directed from lower potential to higher then the direction is taken to be positive. • If the field is directed from higher potential to lower potential then the direction is taken as negative. Electric Field and Electric Potential Relation Test charge Formula Electric gradient Positive \(\begin \) Electric potential is perpendicular to Electric field lines. Electric Field and Electric Potential Relation Derivation Let us study how to find the electric potential of the electric field is given. Work done in moving the test 0 from a to b is given by- \(\begin \) Hope you understood the relation and conversion between Electric Field and Electric potential. Physics Related Topics: Stay tuned with Byju’s for more such interesting articles. Also, register to “BYJU’S-The Learning App” for loads of interactive, engaging physics related videos and an unlimited academic assist.