Informative line

Direction Of Magnetic Field

Learn magnetic field definition with right hand rule magnetic field as cross product of two vectors, practice to calculate direction of magnetic field around a wire & due to current carrying wire and magnetic field of a point charge and a moving charge.

Cross Product of Two Vectors

  • Consider two vectors \(\vec A\) and \(\vec B\) making an angle \(\theta\) with each other.

  • The cross product of \(\vec A\) and \(\vec B\) is defined as

\(\vec A×\vec B=|\vec A|\,|\vec B|sin\theta \,\,\hat n\)

where \(\hat n\) is the unit vector in the direction of \(\vec A×\vec B\).

  • Cross product is also known as vector product as cross product of two vector quantities gives a vector quantity.

Direction of \((\vec A×\vec B)\)

  • The direction of \((\vec A×\vec B)\) is specified by using the right hand thumb rule which states, 

                     Curl the fingers of your right hand from \(\vec A\) towards \(\vec B\) with your thumb                        extending outwards and perpendicular to the plane of \(\vec A\) and \(\vec B\) as shown in figure.

  • The direction of \((\vec A×\vec B)\) is now shown by your thumb.

 

Fingers curl from \(\vec A\) towards \(\vec B\) and thumb pointing towards direction of \((\vec A×\vec B)\).

Cross product of standard unit vectors :

Direction of cross product of two vector quantities can be understood using standard unit vectors.

Three unit vectors in 3-coordinate system, is shown in figure.

  • While moving in the direction of the arrows shown in figure, cross product of any two unit vectors gives the third unit vector.

\(\hat i× \hat j=\hat k\)

\(\hat j× \hat k=\hat i\)

\(\hat k× \hat i=\hat j\)

 

 

  • While moving in the opposite direction of the arrows shown in figure, cross product of any two unit vector gives negative of the third unit vector.
  • \(\hat k× \hat j=-\hat i\)

    \(\hat j× \hat i=-\hat k\)

    \(\hat i× \hat k=-\hat j\)

     

     

  • This can also be understood by,

\(\vec A×\vec B=-(\vec B×\vec A)\)

 

 

Illustration Questions

The direction of \(\vec A×\vec B\) will be

A \(-\hat k\)

B \(\hat k\)

C \(\hat j\)

D \(-\hat j\)

×

According to right hand thumb rule, the direction of \(\vec A×\vec B\) will be perpendicular to the plane of  \((\vec A×\vec B)\).

Since, \((\vec A×\vec B)\) lie in x-y plane. So, the direction of \((\vec A×\vec B)\) will be in the direction of unit vector \(\hat k\).

The direction of \(\vec A×\vec B\) will be

image
A

\(-\hat k\)

.

B

\(\hat k\)

C

\(\hat j\)

D

\(-\hat j\)

Option B is Correct

Direction of Magnetic Field (B) at a Point due to Point Charge

  • Consider a positive point charge moving with some velocity \(\vec v\), as shown in figure.

  • The direction of magnetic field at P will be determined using right hand thumb rule.
  • Place your right hand around the line in such a way that your fingers sweep \(\vec v\) into \(\vec r\) through the smaller angle between them. The outstretched thumb will point in the direction of magnetic field \((\vec B)\).

                                                  OR

  • Point the right thumb in the direction of \(\vec v\). The magnetic field vector \(\vec B\) is perpendicular to the plane of \(\vec r\) and \(\vec v\), pointing in the direction in which fingers curl. 

Case: Point charge going inside the page 

  • To calculate magnetic field for a point charge going inside the page, use right hand thumb rule.

  • Direct your right hand thumb inside the page. Now, curl your fingers, it will give the direction of magnetic field.

Illustration Questions

A positive charge is moving in +y direction with a velocity \( \vec v\). The direction of magnetic field at point P whose z-coordinate is zero, as shown in figure, will be along unit vector -

A \(\hat i\)

B \(-\hat k\)

C \(\hat k\)

D \(\hat j\)

×

Since, \(\vec v\) and \(\vec r\) lies in x-y plane. So, direction of \((\vec v×\vec r)\) will be along \(\hat k\).

image

Hence, the direction of magnetic field will be perpendicular to plane of \((\vec v\;and\;\vec r)\).

So, the direction of magnetic field is along \(\hat k\).

A positive charge is moving in +y direction with a velocity \( \vec v\). The direction of magnetic field at point P whose z-coordinate is zero, as shown in figure, will be along unit vector -

image
A

\(\hat i\)

.

B

\(-\hat k\)

C

\(\hat k\)

D

\(\hat j\)

Option C is Correct

Direction of Magnetic Field for a Negative Charge

  • Consider a negative charge going inside a page.

  • To determine the direction of magnetic field at point P as shown in figure, following steps are to be followed.

Step 1: Find the direction of magnetic field considering negative charge as a positive charge.

Step 2: Opposite the direction of magnetic field.

Illustration Questions

Find the direction of magnetic field at 0 if a negative charge is moving upward from point P (4, 5).

A \(\hat i\)

B \(-\hat k\)

C \(\hat k\)

D \(\hat j\)

×

Direction of magnetic field when positive charge is placed

Since, \(\vec r\) and \(\vec v\) both are in x-y plane. So, direction of magnetic field will be perpendicular to the plane of \(\vec v \,\,\,\)and\(\,\,\,\vec r\) for positive charge (according to right hand thumb rule).

 

So, direction of magnetic field for positive charge = \(\hat k\)

image

For negative charge, direction of magnetic field will be opposite to that of positive charge.

Direction of magnetic field at O = Direction of \((\vec r×\vec v)\) = –direction of \((\vec v×\vec r)\)

\(-\hat k\)

Find the direction of magnetic field at 0 if a negative charge is moving upward from point P (4, 5).

image
A

\(\hat i\)

.

B

\(-\hat k\)

C

\(\hat k\)

D

\(\hat j\)

Option B is Correct

Direction of Magnetic Field due to Current Carrying Wire

  • Consider a current carrying wire as shown in figure.

 

  • To find the direction of magnetic field about this wire, use right hand thumb rule.
  • According to right hand thumb rule, if the right hand thumb is placed in the direction of the current and fingers are curled around the wire representing a circle, then the finger point in the direction of magnetic field around the wire.

Example - 

  • Consider a current carrying wire in which the current is flowing inward.

Illustration Questions

In all the options, a current carrying wire and a point (P) are shown. Choose the option in which the direction of magnetic field is not outwards.

A

B

C

D

×

Direction of magnetic field \((\vec B)\) can be determined by right hand thumb rule.

In option A

Since, fingers curl outward, the direction of magnetic field is outward.

image

In option B

Since, fingers curl outward, the direction of magnetic field is outward.

 

image

In option C

Since, fingers curl outward, the direction of magnetic field is outward.

image

In option D

Since, fingers curl inward, the direction of magnetic field will be inward.

image

In all the options, a current carrying wire and a point (P) are shown. Choose the option in which the direction of magnetic field is not outwards.

A image
B image
C image
D image

Option D is Correct

Magnetic Field due to Small Element of Current Carrying Wire

  • Consider a long, current carrying wire, as shown in figure.
  • To calculate magnetic field due to small element of wire, a small element AB is selected.

  • To determine the direction of magnetic field at a point P which is far away from the element, consider the element of any arbitrary shape to be straight line.

  • The direction of magnetic field at point P can be specified by assuming the wire to be straight line.

  • Applying right hand thumb rule, the direction of magnetic field at P by this wire is outward.

Example :

Direction of magnetic field at P due to small element AB of a current carrying wire loop can be specified by assuming the element as a straight wire.

  • Considering a small element AB of wire.

  • By applying right hand thumb rule, it is concluded that direction of magnetic field is  inward.

Illustration Questions

A current carrying wire is shown in figure. The direction of magnetic field at point A due to small element PQ as shown in figure will be - [Assume the element to be very small than the distance of the point from the element ]

A Upward

B Downward

C Inwards

D Outwards

×

Considering a small element PQ of wire to calculate electric field at P.

image

Since the wire is very small it can be considered as a straight line.

image

According to right hand thumb rule,

The direction of magnetic field at P will be in outward direction.

image

A current carrying wire is shown in figure. The direction of magnetic field at point A due to small element PQ as shown in figure will be - [Assume the element to be very small than the distance of the point from the element ]

image
A

Upward

.

B

Downward

C

Inwards

D

Outwards

Option D is Correct

Magnetic Field at the Axis of a Current Carrying Loop due to Small Section of Wire

  • Consider a wire loop placed in y-z plane such that it is symmetrical along x-axis.

  • Consider an element ds on this wire loop and a point P on the x-axis as shown in figure.

  • Magnetic field at point P due to small element ds can be specified by considering the wire as straight line.
  • By seeing it from front the direction of current in ds is outward.

  • By applying right hand thumb rule, the direction of magnetic field at point P is shown in figure.

Illustration Questions

A rectangular wire loop carrying a current in the direction  as shown in figure is placed in x-z plane. The direction of magnetic field at point P due to side CD of the rectangular loop will be along unit vector

A \(\hat i\)

B \(\hat j\)

C \(-\hat k\)

D \(-\hat j\)

×

By applying right hand thumb rule, the direction of magnetic field will be along \(-\hat j\).

image

Front view of arm CD

image

A rectangular wire loop carrying a current in the direction  as shown in figure is placed in x-z plane. The direction of magnetic field at point P due to side CD of the rectangular loop will be along unit vector

image
A

\(\hat i\)

.

B

\(\hat j\)

C

\(-\hat k\)

D

\(-\hat j\)

Option D is Correct

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