In the $$xy$$-plane, the region $$y > 0$$ has a uniform magnetic field $${B_1}\widehat k$$ and the region $$y < 0$$ has another uniform magnetic field $${B_2}\widehat k.$$ A positively charged particle is projected from the origin along the positive $$y$$-axis with speed $${v_0} = \pi \,m{s^{ - 1}}$$ at $$t=0,$$ as shown in the figure. Neglect gravity in this problem. Let $$t=T$$ be the time when the particle crosses the $$x$$-axis from below for the first time. If $${B_2} = 4{B_1},$$ the average speed of the particle, in $$m{s^{ - 1}},$$ along the $$x$$-axis in the time interval $$T$$ is ___________.

In electromagnetic theory, the electric and magnetic phenomena are related to each other. Therefore, the dimensions of electric and magnetic quantities must also be related to each other. In the questions below, $$[E]$$ and $$[B]$$ stand for dimensions of electric and magnetic fields respectively, while $$\left[ {{\varepsilon _0}} \right]$$ and $$\left[ {{\mu _0}} \right]$$ stand for dimensions of the permittivity and permeability of free space respectively. $$\left[ L \right]$$ and $$\left[ T \right]$$ are dimensions of length and time respectively. All the quantities are given in $$SI$$ units.

The relation between $$\left[ {{\varepsilon _0}} \right]$$ and $$\left[ {{\mu _0}} \right]$$ is

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $$z = x/y.$$ If the errors in $$x,y$$ and $$z$$ are $$\Delta x,\Delta y$$ and $$\Delta z,$$ respectively, then
$$$z \pm \Delta z = {{x \pm \Delta x} \over {y \pm \Delta y}} = {x \over y}\left( {1 \pm {{\Delta x} \over x}} \right){\left( {1 \pm {{\Delta y} \over y}} \right)^{ - 1}}.$$$

The series expansion for $${\left( {1 \pm {{\Delta y} \over y}} \right)^{ - 1}},$$ to first power in $$\Delta y/y.$$ is $$1 \pm \left( {\Delta y/y} \right).$$ The relative errors in independent variables are always added. So the error in $$z$$ will be
$$$\Delta z = z\left( {{{\Delta x} \over x} + {{\Delta y} \over y}} \right).$$$

The above derivation makes the assumption that $$\Delta x/x < < 1,$$ $$\Delta y/y < < 1.$$ Therefore, the higher powers of these quantities are neglected.

In an experiment the initial number of radioactive nuclei is $$3000.$$ It is found that $$1000 \pm 40$$ nuclei decayed in the first $$1.0s.$$ For $$\left| x \right| < < 1.$$ $$\ln \left( {1 + x} \right) = x$$ up to first power in $$x.$$ The error $$\Delta \lambda ,$$ in the determination of the decay constant $$\lambda ,$$ in $${s^{ - 1}},$$ is

If the measurement errors in all the independent quantities are known, then it is possible to determine the error in any dependent quantity. This is done by the use of series expansion and truncating the expansion at the first power of the error. For example, consider the relation $$z = x/y.$$ If the errors in $$x,y$$ and $$z$$ are $$\Delta x,\Delta y$$ and $$\Delta z,$$ respectively, then
$$$z \pm \Delta z = {{x \pm \Delta x} \over {y \pm \Delta y}} = {x \over y}\left( {1 \pm {{\Delta x} \over x}} \right){\left( {1 \pm {{\Delta y} \over y}} \right)^{ - 1}}.$$$

The series expansion for $${\left( {1 \pm {{\Delta y} \over y}} \right)^{ - 1}},$$ to first power in $$\Delta y/y.$$ is $$1 \pm \left( {\Delta y/y} \right).$$ The relative errors in independent variables are always added. So the error in $$z$$ will be
$$$\Delta z = z\left( {{{\Delta x} \over x} + {{\Delta y} \over y}} \right).$$$

The above derivation makes the assumption that $$\Delta x/x < < 1,$$ $$\Delta y/y < < 1.$$ Therefore, the higher powers of these quantities are neglected.

Consider the ratio $$r = {{\left( {1 - a} \right)} \over {1 + a}}$$ to be determined by measuring a dimensionless quantity $$a.$$ If the error in the measurement of $$a$$ is $$\Delta a\left( {\Delta a/a < < 1.} \right.$$ then what is the error $$\Delta r$$ in determining $$r$$?