Learn magnetic flux definition with equation, practice to calculate the total magnetic flux through the loop due to current in the wire and find electric flux through cylinder and magnetic field of a current loop.

- Flux of any physical quantity is the measure of flow of that quantity passing through any specific area.
- To understand the idea of flux, consider a region in which rain is falling vertically downwards and ring is placed in this region.
- The rate of flow of rain passing through area of the ring depends on its area and orientation of ring.

Since,

Area c > Area b > Area a .

Hence, flux through

c > b > a

- Plane of ring (a) oriented perpendicular to field.
- Plane of ring (b) oriented at an angle to the normal of the field.
- Plane of ring (c) oriented along the field.
- Flux through Ring (a) > Ring (b) > Ring (c).
- Flux of Ring (c) is zero as its plane is oriented along the field so, no rain is passing through its area.

Magnetic flux is defined as total number of magnetic lines passing through a specified area placed in a magnetic field.

The SI unit of magnetic flux is Weber (Wb) or T.m^{2} (Tesla.m^{2}).

A a > b > c

B a = b = c

C c > b > a

D a > c > b

- Consider a closed surface of cylinder placed in a non-uniform magnetic field as shown in figure.
- Area of P = Area of Q.
- Number of field lines associated with P is greater than number of field lines associated with Q.
- Flux linked with P is greater than flux linked with Q.
- Flux is denoted by \(\phi\).

\(\phi_P>\phi_Q\)

- Consider a body placed in a magnetic field.
- Calculate number of field lines entering and leaving the body.

**Case 1**: If the number of field lines entering and leaving the body are equal, then total flux linked with the body will be zero.

**Case 2**: If the number of field lines entering and leaving the body are not equal, then

Net flux = Number of field lines leaving - Number of field lines entering

- When any body is placed in an uniform magnetic field such that its surface is perpendicular to the magnetic field, then magnetic flux associated with it, is maximum.
- Now if, surface of same body is placed at some angle with the magnetic field rather than perpendicular, then magnetic flux associated with it, will be less than maximum value.
- The number of magnetic field lines passing through the tilted surface is less than the number of field lines passing through perpendicular surface but same as the number of field lines passing through the projection of area of surface.

Number of field lines passing through tilted surface (A) = Number of field lines passing through projection of area of surface

\(\phi_{m'_\bot}=\phi_{m'}<\phi_m\)

\(A_{m'_\bot} < A_{m'} = A_m\)

A \(\phi_A>\phi_B\)

B \(\phi_B>\phi_A\)

C \(\phi_A=\phi_B\)

D

- Consider a cylindrical body placed in a uniform magnetic field \(\vec B\).
- To calculate flux, by convention area vector is always taken perpendicular and in outward direction to the surface.

- Area vector of three surfaces of a cylindrical body, are shown in figure.

- By convention, area vector is always taken perpendicular and in outward direction to the surface.
- Angle between \(\vec B\) and \(d\vec A_1\) is 180°.

\(\phi_1=\vec B\cdot d\vec A_1\)

\(\phi_1=B\;dA_1\;cos(180°)\)

\(\phi_1=-B\ dA_1\)

- It can be concluded that for surface 1 by convention, we have taken the \(d\vec A\) outside the surface which is in opposite direction to magnetic field.
- Hence, the flux coming out of surface 1 is negative.

**Conclusion:** For any particular surface through which the magnetic field lines are entering, flux will be negative.

Angle between \(\vec B\) and \(d\vec A_2\) is 90°.

\(\phi_2=\vec B\cdot d\vec A_2\)

or, \(\phi_2=B\;dA_2\;cos(90°)\)

or, \(\phi_2=0\)

**Conclusion:** If magnetic field and area vector are perpendicular to each other, then flux through that surface is zero.

Angle between \(\vec B\) and \(d\vec A\) is 0°.

\(\phi_1=\vec B\cdot d\vec A\)

\(\phi_1=B\;dA\;cos0°\)

\(\phi_1=B\cdot dA\)

- By convention, for surface 3 we have taken \(d\vec A\) outside the surface which is in the direction of magnetic field.
- Hence, flux coming out of surface 3 is positive.

**Conclusion:** For any particular surface through which the magnetic field lines are leaving, flux will be positive.

A \(B_zA_1,\,-B_xA_2,\,+B_zA_3\)

B \(B_zA_1,\,B_xA_2,\,-B_zA_3\)

C \(B_zA_1,\,-B_xA_2,\,-B_zA_3\)

D \(-B_zA_1,\,B_xA_2,\,B_zA_3\)

- The magnetic field due to current carrying wire is non-uniform.
- The magnetic field at a point near the wire is more.

Magnetic flux through a current carrying loop is given as

\(\phi _m=\int\vec B.d\vec A\)

where \(\vec B\) = magnetic field

\(d\vec A\) = Area vector

\(\phi _m\) = magnetic flux

A \(\mu_0Ia\,\ell n\left(1+\dfrac{a}{d}\right)\)

B \(\dfrac{\mu_0Ia}{2\pi}\,\ell n\left(1+\dfrac{a}{d}\right)\)

C \(\mu_02\pi\,\ell n(Ia)\)

D \(\dfrac{\mu_02\pi}{a}\,\ell n\left(1+\dfrac{a}{d}\right)\)

- By convention, area vector is always taken outwards / perpendicular to the plane, as shown in figure.
- Consider a closed surface, as shown in figure. Area vector for all the three surfaces A, B and C of cylinder is taken outwards/perpendicular to the surface.

Consider a ring placed in a uniform magnetic field, as shown in figure.

- Magnetic flux linked with the area A is given as the dot product of magnetic field and area vector.
- Flux linked with area A

\(\phi_B=B(A\;cos\theta)\)

\(\phi_B=\vec B\cdot\vec A\)

- For non-uniform magnetic field, magnetic field may vary over a large surface.
- The concept of flux is meaningful for small area where magnetic field is approximately constant, when total surface is placed in non-uniform field.

\(\Delta\phi_B=B\cdot(\Delta A\;cos\theta)\)

or, \(\Delta\phi_B=\vec B\cdot\Delta \vec A\)

where \(\Delta\phi_B\) is the magnetic field through this element

- The magnetic flux through whole surface is given as summation of magnetic flux through all these small elements.

\(\phi_B=\sum\vec B\cdot\Delta\vec A\)

- If area of each element approaches zero, the number of elements approach infinity, then the sum is replaced by integral.

\(\phi_B=\int_s\vec B\cdot d\vec A\)

- For closed surface, \(\phi_B=\int\vec B\cdot d\vec A\)
- If in a region magnetic field is given as \(\vec B=B_x\hat i+B_y\hat j+B_z\hat k\)
- An area A is placed in an magnetic field with area vector \(\vec A=A_x\hat i+A_y\hat j+A_z\hat k\)
- Flux linked with this area is given as \(\phi=\vec B\cdot\vec A\)

\(\phi=B_xA_x+B_yA_y+B_zA_z\)

- By convention, area vector is always taken outwards / perpendicular to the plane, as shown in figure.
- Consider a closed surface, as shown in figure. Area vector for all the three surface A, B and C of cylinder is taken outwards/perpendicular to the surface.

Consider a ring placed in a uniform magnetic field, as shown in figure.

- Magnetic flux linked with the area A is given as the dot product of magnetic field and area vector.
- Flux linked with area A

\(\phi_B=B(A\;cos\theta)\)

\(\phi_B=\vec B\cdot\vec A\)