Problem 53 - Master Resource Book in JEE Main Physics

Question

The dimensions of magnetic field in \(M, \mathrm{~L}, \mathrm{~T}\) and \(\mathrm{C}\) (Coulomb) are given as [AIEEE 2008] (a) \(\left[\mathrm{MLT}^{-1} \mathrm{C}^{-1}\right]\) (b) \(\left[\mathrm{MT}^{2} \mathrm{C}^{-2}\right]\) (c) \(\left[\mathrm{MT}^{-1} \mathrm{C}^{-1}\right]\) (d) \(\left[\mathrm{MT}^{-2} \mathrm{C}^{-1}\right]\)

Step-by-Step Solution

Answer

The dimensions of the magnetic field are \([ ext{MT}^{-1} ext{C}^{-1}]\), which corresponds to option (c).

Step 1: Understand the Magnetic Field

The magnetic field is a fundamental vector field described in physics that influences magnetic materials and moving charges. In SI units, it is expressed in teslas (T). The relationship is defined via the formula: \[ F = q(v \times B) \] where \( F \) is the magnetic force, \( q \) is the charge, \( v \) is the velocity, and \( B \) is the magnetic field.

Step 2: Analyze Force Dimensionally

Let's first write the dimensions of force \( F \) as \( [ ext{MLT}^{-2}] \), where M is mass, L is length, and T is time. This is important since it gives us the dimensions to relate to the magnetic field.

Step 3: Write the Dimensions of Related Quantities

Now, analyze the other terms:- Charge \( q \) has a dimension of \( [ ext{C}] \).- Velocity \( v \) has a dimension of \( [ ext{LT}^{-1}] \).Therefore, \( v \times B \) must equal force when multiplied by charge.

Step 4: Derive the Dimensions of Magnetic Field

We rearrange the formula \( F = q(v \times B) \) to isolate the magnetic field dimensions: \[ B = \frac{F}{qv} \]Substitute the dimensions:\[ B = \frac{[ ext{MLT}^{-2}]}{[ ext{C}][ ext{LT}^{-1}]} \]Simplifying this expression, we get:\[ B = [ ext{MT}^{-1} ext{C}^{-1}] \]

Step 5: Choose the Correct Option

Compare the derived dimensional formula \([ ext{MT}^{-1} ext{C}^{-1}]\) with the given options. The correct match is (c) \([ ext{MT}^{-1} ext{C}^{-1}]\).

Key Concepts

Magnetic Field Dimensions

When talking about magnetic fields in physics, dimensions play a crucial role in understanding how the field interacts with charges and currents. In this context, dimensions refer to the base quantities like mass (M), length (L), time (T), and charge (Coulomb, C), which are combined to express the dimensions of various physical quantities. To determine the dimensions of a magnetic field, we use the relationship that involves the magnetic force, velocity, and charge.

From the formula \(F = q(v \times B)\), the dimensions of magnetic field \(B\) are derived by rearranging terms and substituting in their respective dimensions. Here's a step-by-step breakdown:

• The dimension of magnetic force \(F\) is \([MLT^{-2}]\).
• Charge \(q\)'s dimension is \([C]\).
• Velocity \(v\) is expressed dimensionally as \([LT^{-1}]\).
• Thus, the dimensional formula for \(B\) becomes \([MT^{-1}C^{-1}]\).

This dimensional expression allows us to understand and predict how the magnetic field would interact with different physical quantities.

Magnetic Force Formula

The magnetic force formula is central to understanding how magnetic fields influence charges and currents. It's given by the equation \(F = q(v \times B)\), wherein each variable plays a specific role:

• \(F\) represents the magnetic force exerted.
• \(q\) is the charge that experiences the force.
• \(v\) denotes the velocity of the charge moving within the magnetic field.
• \(B\) is the magnetic field itself.

This equation describes the magnitude and direction of the force acting on a charged particle moving through a magnetic field. What's particularly interesting about this formula is that the force is perpendicular to both the velocity of the charge and the direction of the magnetic field. This perpendicularity is what causes the charge to move in a circular or spiral path rather than a straight line. Moreover, it's essential to recognize that this formula leads us to the dimensional analysis of the magnetic field.

SI Units for Magnetic Field

In the International System of Units (SI), the magnetic field is measured in teslas (T). This unit is named after the inventor Nikola Tesla, whose work contributed significantly to the understanding of electromagnetic fields. The tesla is defined as a derived SI unit of measure. Here's how it's connected to other SI units:

• One tesla is equivalent to one weber per square meter (T = Wb/m²).
• It also relates to newtons, as 1 T is the force of 1 newton on a 1 coulomb charge moving at 1 meter per second perpendicular to the magnetic field.
• The units reflect dimensions \([MT^{-1}C^{-1}]\), derived from the force equation.

Understanding these units is critical because it provides a standardized way to measure and calculate magnetic fields across various physics problems and applications. It simplifies communication and understanding among scientists and engineers globally.