Gradient of scalar field.pptx

The gradient of a scalar field, the Physical significance of the gradient, and numerical problems on the gradient of a scalar field
for B.Sc Physics - Mechanics - first year first -semester

1. B.Raju Dept.of Physics., KGC Hanamkonda Vector Analysis Lecture - 2 B.Sc Physics - First Year - First Semester Topic: Gradient of Scalar Field and Numericals on gradient

2. Dept.of Physics., KGC Hanamkonda ∇ Gradient of Scalar field operated on Scalar field( Φ ) Vector field ( ∇ Φ ) Operating del operator on a scalar field results in gradient and the gradient of a scalar field gives the maximum rate of change of the field at a point

3. Dept.of Physics., KGC Hanamkonda Φ Level Surface Normal direction Level Surface ( Equipotential surface): A surface on which the value of the field is constant (refers to a  region  in space where every point in it is at the same value.)

4. Dept.of Physics., KGC Hanamkonda Gradient of a scalar field: Let two level surfaces ( equipotential surfaces) S 1 and S 2 very close to each other. The level surfaces are specified by scalar values S and S + dS . Take points A and B on S 1 and S 2 respectively and O is (a point)origin outside the surfaces as shown in figure. Let r and r + dr be the radius vectors of points A and B with respect to origin O respectively.

5. Dept.of Physics., KGC Hanamkonda The vector drawn from A to B is The least distance between S 1 and S 2 is AC. The direction of AC is along is unit normal vector the length of AC is dn The rate of change of S at A in the direction AC =

6. But, gives the maximum rate of change at A along the direction of AC. This quantity is called gradient. ∴ Therefore, the gradient of a scalar quantity is a vector ∴ gradS.dr = ds -----------(4)

7. Dept.of Physics., KGC Hanamkonda But, in cartesian coordinate system the scalar function S is taken as S = S(x,y,z)

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9. Dept.of Physics., KGC Hanamkonda Physical Significance of Gradient By operating Del operator on scalar field we can get vector field. Gradient of a scalar field gives the maximum rate of change of field at a point(gradient of a scalar point function represents normal vector to the level surface .) Vector fields obtained by gradient of scalar fields are called as Lamellar fields eg : Electric field intensity E = - ∇ V Gradient of gravitational potential gives gravitational field intensity In Lamellar fields, the line integral over a closed curve is always zero Gradient is independent of coordinate system, so gradient is invariant

10. Problem1 .If the potential of a scalar field is . Find the gradient of the field. Solution: grad φ = ? grad φ = 𝛁 φ = 𝛁 ( )

11. Dept.of Physics., KGC Hanamkonda Problem 2

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15. Dept.of Physics., KGC Hanamkonda Assignment: 1.Define gradient of a scalar field and write its physical significance 2.Derive an expression for Gradient of a scalar field 3.What are Lamellar fields, give examples for lamellar fields.

16. Dept.of Physics., KGC Hanamkonda Numericals :

17. Dept.of Physics., KGC Hanamkonda Thank you